Chapter 7
Probability
Probability as a general concept can be defined as the chance of an event occuring. Many people are famaliar with Probability from observation or playing games of chance, such as card games, slot machines of lotteries. In addition to being used in games of chance, Probability theory is used in the fields of insurance, investments, weather forecasting and in various other areas.Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665) laid the foundation of Probability theory.
Probability၏ယေဘုယျအယူအဆမှာဖြစ်ရပ်တစ်ခုထွက်ပေါ်ခြင်းအခွင့်အလမ်းဟုသတ်မှတ်နိုင်သည်။လူအများစုသည်အခွင့်အလမ်းရှိသည့်ကစားနည်းများ(ဖဲရိုက်ကစားခြင်း၊အကြွေစေ့ထည့်စက်များ၊ထီများ)အစရှိသည့်ကစားနည်းများမှအတွေ့အကြုံများအရ'ဖြစ်တန်စွမ်း'နှင့်ရင်းနှီးကျွမ်းဝင်ပြီးဖြစ်သည်။ဤအခွင့်အလမ်းရှိသည့်ကစားနည်းများအပြင်အာမခံလုပ်ငန်း၊ရင်းနှီးမြုပ်နှံမှုလုပ်ငန်း၊ရာသီဉတုခန့်မှန်းခြင်းနှင့်များစွာသောနယ်ပယ်တို့တွင်'ဖြစ်တန်စွမ်းသီအိုရီ'ကိုအသုံးပြုလာကြသည်။Blaise Pascal(1623-1662)နှင့်Pierre de Fermat(1601-1665)တို့က'ဖြစ်တန်စွမ်းသီအိုရီ'၏အခြေခံကိုချမှတ်ခဲ့ကြသည်။
For the experiment "tossing a coin", there are two possible outcomes: head and tail (which will be denoted by H snd T respectively). The sample space is $\{H,T\}$, and the events are $\phi$, $\{H\}$, $\{T\}$ and $\{H,T\}$. But for the experiment "tossing two coins", the sample space is $\{(H,H), (H,T), (T,H), (T,T)\}$, which can also be expressed as $\{HH,HT, TH, TT\}$. The event $\{(H,T)\}$ means that "the first toss shows head and the second toss shows tail."
အကြွေစေ့တစ်စေ့မြှောက်သည့် စမ်းသပ်ချက်အတွက် ဖြစ်နိုင်သမျှသောရလဒ်နှစ်ခု Head(H) နှင့် Tail(T) ရှိသည်။
ထို့ကြောင့် the sample space $=\{H,T\}$
events များမှာ $\phi$, $\{H\}$, $\{T\}$ နှင့် $\{H,T\}$
အကြွေစေ့နှစ်စေ့မြှောက်သည့် စမ်းသပ်ချက်အတွက်
the set of sample space ={(H,H), (H,T), (T,H), (T,T)}
{HH,HT, TH, TT} ဟုလည်းဖော်ပြနိုင်သည်။
The event $\{(H,T)\}$ ၏အဓိပ္ပါယ်မှာ ပထမအကြွေစေ့မှ head ကျပြီး၊ဒုတိယအကြွေစေ့မှ tail ကျသည့်ဖြစ်ရပ်ဟုဆိုလိုသည်။
အကြွေစေ့နှစ်စေ့ မြှောက်သည့်စမ်းသပ်ချက်တွက် event အရေအတွက် $16$ ရှိသည်။
For the experiment "rolling a die", there are six possible outcomes: $1$, $2$, $3$, $4$, $5$, $6$. The sample space is $\{1, 2, 3, 4, 5, 6\}$. Some of the events are $\{3\}$, $\{4, 5, 6\}$ and $\{2, 3, 5\}$, which can respectively be described as "the result is $\{3\}$", "the result is at least $\{4\}$" and "the result is a prime number".
အန်စာတုံးတစ်တုံးလှိမ့်သည့်စမ်းသပ်ချက်အတွက် ဖြစ်နိုင်သမျှသောရလဒ်ခြောက်ခု ($1$, $2$, $3$, $4$, $5$, $6$) ရှိသည်။
The sample space $=\{1, 2, 3, 4, 5, 6\}$
events အချို့မှာ $\{3\}$, $\{4, 5, 6\}$ and $\{2, 3, 5\}$ စသဖြင့်။
$\{3\}$ 'the result is $3$' (ရလဒ် $3$ ဖြစ်သည်။)
$\{4, 5, 6\}$ 'the result is at least $4$' (ရလဒ်အနည်းဆုံး $4$ ဖြစ်သည်။)
$\{2, 3, 5\}$ 'the result is a prime' (ရလဒ်သည် သုဒ္ဓကိန်းတစ်ခုဖြစ်သည်။)
The following table represents the sample space for rolling two dice. The first part of an ordered pair in the table represents the number appears on the first die and the second part represents the corresponding number on the second die. Sample spaces of similar experiments can also be obtained by constructing such tables.
အောက်ပါဇယားသည် အန်စာတုံးနှစ်တုံးလှိမ့်သည့်စမ်းသပ်ချက်အတွက် sample space များကိုဖော်ပြသည်။ ordered pair တစ်ခုစီ၏ပထမကိန်းသည် ပထမအန်စာတုံးပေါ်မှကိန်းများကိုကိုယ်စားပြုပြီး၊ဒုတိယကိန်းသည်ဒုတိယအန်စာတုံးပေါ်မှသက်ဆိုင်ရာကိန်းများကိုကိုယ်စားပြုသည်။အလားတူစမ်းသပ်ချက်များအတွက်ဤကဲ့သို့သော table များပြု လုပ်၍ sample space များကိုရရှိနိုင်သည်။
Sometimes, it is convenient to use a tree diagram to list all possible outcomes in a sample space. The following tree diagram displays the sample space for tossing a coin three times.
တစ်ခါတရံ tree diagram ကိုအသုံးပြု၍ sample space တစ်ခုအတွင်းရှိ ဖြစ်နိုင်သမျှသောရရဒ်များအားလုံးကိုစီစဉ်နိုင်သည်။အောက်ပါ tree diagram သည်အကြွေစေ့တစ်စေ့ကိုသုံးကြိမ်မြှောက်ခြင်းစမ်းသပ်ချက်မှ sample space ကိုပြသနေသည်။
Example $1$
Find the probability of randomly selecting a red pen from a box that contains $2$ red pens, $4$ blue pens and $3$ yellow pens.
Example $2$
If a whole number from $1$ to $20$ both inclusive is randomly selected, and if each number has an equal chance of being selected, what is the probability that the number will be
(a) even? (b) greater than $1$ (c) prime?
Example $3$
In a sample of $50$ people, $21$ had type O blood, $22$ had type A blood, $5$ had type B blood and,
$2$ had type AB blood. Find the following probabilities.
(a) A person has type O blood.
(b) A person has type A or type B blood.
(c) A person has neither type A nor type O blood.
(d) A person does not have type AB blood.
$\text{ (a) P(O) }=\dfrac{21}{50}$
$\text{ (b) P(A or B) }=\dfrac{27}{50}$
$\text{ (c) P(neither A nor O) }=\dfrac{7}{20}$
$\text{ (d) P(not AB) }=\dfrac{48}{50}=\dfrac{24}{25}$
Example $4$
A family has three children. Find the probabilities of(a) all boys, (b) exactly two boys,
(c) at most two boys, (d) at least one girl,
(e) at least one boy and at least one girl.
$\text{ The sample space consists of } 8 \text{ outcomes. }$
$\text{ (a) The event is }\{BBB\}$
$\text{ P(all boys) }=\dfrac{1}{8}$
$\text{ (b) The event is }\{BBG, BGB, GBB\}$
$\text{ P(exactly two boys) }=\dfrac{3}{8}$
$\text{ (c) The event is }\{BBG, BGB, BGG, GBB, GBG, GGB, GGG\}$
$\text{ P(at most two boys) }=\dfrac{7}{8}$
$\text{ (d) The event is }\{BBG, BGB, BGG, GBB, GBG, GGB, GGG\}$
$\text{ P(at least one girl) }=\dfrac{7}{8}$ (or)
$\text{ P(at least one girl) } =1- \text{ P(all boys) }=1-\dfrac{1}{8}=\dfrac{7}{8}$
$\text{ (by using rule for complementary events) }$
$\text{ (e) The outcomes is }\{BBG, BGB, BGG, GBB, GBG, GGB \}$
$\text{ P(at least one boy and at least one girl) }=\dfrac{6}{8}=\dfrac{3}{4}$
Example $5$
Draw a tree diagram to list all possible two-digit numerals which can be formed by using the digits $2$, $3$, $5$ and $6$ without repeating any digit. If one of these numerals is chosen at random, find the probability that it is divisible by $13$. Find also the probability that it is either a prime or a perfect square. Find the probability that none of its digits is $6$.
$\text{ The sample space consists of }12 \text{ outcomes. }$
$\text{ The event containing numerals divisible by }13 \text{ is }\{26, 52, 65\}$
$\text{ P(a numeral which is divisible by }13 \text{ ) }=\dfrac{3}{12}=\dfrac{1}{4}$
$\text{ The event containing numerals which are primes or perfect square is }\{23, 25, 36, 53\}$
$\text{ P(a numeral which is prime or a perfect square) }=\dfrac{4}{12}=\dfrac{1}{3}$
$\text{ The event containing numerals which do not have digit }6 \text{ is }\{23, 25, 32, 35, 52, 53\}$
$\text{ P(a numeral none of its digits is $6$) }=\dfrac{6}{12}=\dfrac{1}{2}$
Example $6$
Two fair dice are thrown and the numbers appeared on top faces are recorded.Find the probability of each event:
(a) The first die shows $5$.
(b) The sum of the numbers on the dice is $7$.
(c) The product of the numbers on the two dice is greater than $24$.
Example $7$
A bag contains $3$ red balls, $2$ blue balls and $5$ white balls. A ball is selected at random and its colour noted. Then it is replaced. A second ball is selected and its colour noted. Find the probabilities of : (a) selecting $2$ blue balls,(b) selecting $1$ blue ball and then $1$ white ball,
(c) selecting $1$ red ball and then and then $1$ blue ball.
Example $8$
A bag contains $9$ red marbles and $3$ green marbles. For each case belo, find the probability of randomly selecting a red marble on the first draw and a green marble on the second draw.(a) The first marble is replaced. (b) The first marble is not replaced.
Example $9$
At a teachers' conference, there are $4$ English teachers, $3$ Mathematics teachers, and $5$ Science teachers.If $4$ teachers are selected for a committee, find the probability that at least one is a science teacher.Example $10$
In a hospital unit, there are $8$ nurses and $5$ doctors. Among them, there are $7$ nurses and $3$ doctors are females. If a staff person is selected, find the probability that the staff is a nurse or a male.
$\begin{aligned}
\text{ P(nurse or male) }&= \text{ P(nurse)} + \text{ P(male)} - \text{ P(male nurse)}\\
&=\dfrac{8}{13} + \dfrac{3}{13} - \dfrac{1}{13}\\
&=\dfrac{10}{13}
\end{aligned}$
Example $11$
A box contains $3$ strawberry doughnuts, $4$ jelly doughnuts, and $5$ chocolate doughnuts. If a doughnut is selected at random, find the probability that it is either a strawberry doughnut or a chocolate doughnut.Example $12$
Traffic analysis found that the probability that a motorist will turn right at the intersection is $\dfrac{1}{3}$. Out of $300$ motorists, how many would you expect to turn right at that intersection?Example $13$
A spinner is equally to point to any one of the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$. What is the probability of scoring a number divisible by $3$? If the arrow is spun $700$ times, how many would you expect a number not divisible by $3$?Problems
Exercise $7.1$
- A letter is chosen at random from the letters of the word ORANGE.
What is the probability that it is a vowel? - A bag contains $10$ red balls and $30$ black balls. (a) If a ball is drawn at random, what is the probability of getting a red ball?
- How many three-digit numerals can be formed from $1$, $5$ and $7$, without repeating any digit?
Find the probability of a numeral which begins with $1$? - A box contains five cards numbered as $2$, $3$, $4$, $5$ amd $9$. A card is chosen, the number is recorded, and the card is not replaced. Then another card is chosen and the number is recorded. Draw a tree diagram to get the possible outcomes. Find the probabilities of (a) getting two prime numbers, (b) getting two odd numbers and (c) getting a pair of numbers whose sum is a prime number.
- A box contains four marbles of two blue, one red and one yellow. A marble is chosen, the colour is recorded, and the marble is not replaced. Then another marble is chosen and the colour is recorded. Draw a tree diagram to determine possible outcomes. Hence, find the probabilities of (a) getting two blue marbles and (b) getting two different colours.
- A spinner is equally likely to point to any one of the numbers $2$, $3$, $4$ and $5$. Make a table of ordered pairs (first spin, second spin).Find the probability of (a) two odd numbers, (b) an even number followed by an odd number.
- A coin is tossed and then a die is thrown. Head or tail and the number turns up are recorded each time. Draw a tree diagram and list the possible outcomes. Hence, find the probability that head and $6$ turns up.
(b) Suppose the first ball drawn at random is red and is not replaced. If another ball is drawn at random, what is the probability that it will again be red?
%203a.png)
$\text{ The sample space } S \text{ contains } 6 \text{ outcomes. }$%204a.png)
$\text{ The samples space contains } 20 \text{ outcomes. }$
$\therefore \text{ n( } S \text{ ) }=20$
$\text{ (a)The event } E_1 \text{ is }\{(2,3), (2,5), (3,2), (3,5), (5,2), (5,3)\}$
$\therefore \text{ n( }E_1 \text{ ) }=6$
$\text{ P(getting two prime numbers) }=\dfrac{\text{n}(E_1)}{\text{n}(S)} = \dfrac{6}{20}=\dfrac{3}{10}$
$\text{ (b)The event $E_2$ is }\{(3,5), (3,9), (5,3), (5,9), (9,3), (9,5)\}$
$\therefore \text{ n( }E_2 \text{ ) }=6$
$\text{ P(getting two odd numbers) }=\dfrac{ \text{ n }(E_2) }{\text{ n }(S) } = \dfrac{6}{20}=\dfrac{3}{10}$
$\text{ (c) The event } E_3 \text{ is } \{(2,3), (2,5), (2,9), (3,2), (3,4), (4,3), (5,2), (9,2), (9,4), (4,9)\}$
$\therefore \text{ n }(E_3)=10$
$\text{ P(getting a pair of numbers whose sum is a prime number) }=\dfrac{\text{n}(E_3)}{\text{n}(S)} = \dfrac{10}{20}=\dfrac{1}{2}$
%205a.png)
$\text{ The samples space contains } 12 \text{ outcomes. }$
$\therefore \text{ n( } S \text{ ) }=12$
$\text{ (a) The event } E_1 \text{ is }\{(B_1,B_2), (B_2,B_1)\}$
$\therefore \text{ n( } E_1\text{ ) }=2$
$\text{ P(getting two blue marbles) }=\dfrac{\text{ n }(E_1)}{\text{ n }(S)} = \dfrac{2}{12}=\dfrac{1}{6}$
$\text{ (b) The event } E_2 \text{ is } \{(B_1,R), (B_1,Y), (B_2,R), (B_2,Y), (R,B_1), (R,B_2), (R,Y), (Y,B_1), (Y,B_2), (Y,R)\}$
$\therefore \text{ n( } E_2 \text{ ) }=10$
$\text{ P(getting two different marbles) }=\dfrac{\text{ n }(E_2)}{\text{ n }(S)} = \dfrac{10}{12}=\dfrac{5}{6}$
%206a.png)
$\text{ The samples space contains } 16 \text{ outcomes. }$
$\therefore \text{ n( } S \text{ ) }=16$
$\text{ (a) For two odd numbers, }$
$\text{ the event } E_1 \text{ is } \{(3,3), (3,5), (5,3), (5,5)\}$
$\therefore \text{ n }(E_1)=4$
$\text{ P(two odd numbers) }=\dfrac{\text{n}(E_1)}{\text{n}(S)} = \dfrac{4}{16}=\dfrac{1}{4}$
$\text{ (b) For (even, odd) outcomes, }$
$\text{ the event } E_2 \text{ is } \{(2,3), (2,5), (4,3), (4,5)\}$
$\therefore \text{ n }(E_2)=4$
$\text{ P(an even number followed by an odd number) }=\dfrac{\text{n}(E_2)}{\text{n}(S)} = \dfrac{4}{16}=\dfrac{1}{4}$
%207a.png)
$\text{ The samples space contains } 12 \text{ outcomes. }$
$\therefore \text{ n( } S\text{ ) }=12$
$\text{ the event } E \text{ is }\{(H,6)\}$
$\therefore \text{ n( } E \text{ ) }=1$
$\text{ P(head and } 6 \text{ turns up })=\dfrac{\text{n}(E)}{\text{n}(S)} = \dfrac{1}{12}$
Exercise $7.2$
- At a conference, there are $7$ Mathematics instructors, $5$ computer science instructors, $3$ stastics instructors and $4$ science instructors. If an instructor is selected, find the probability of getting a science instructor or a math instructor.
- Two dices are rolled. Find the probability of getting (a) a sum greater than $8$ or a sum less than $3$. (b) a product greater than $9$ or a product less than $16$.
- A bag contains $15$ discs of which $3$ are white, $5$ are red and $7$ are blue. Two discs are to be drawn at random, in succession, each being replaced after its colour has been noted. Calculate the probability that the two discs will be of the same colour.
- In a survey about a change in public policy, $100$ people were asked if they favour
the change, oppose the change, or have no opinion about the change. The responses are
indicated as below.
%204a.png)
Find the probability that a randomly selected respondent to this survey oppose or
has no oponion about the change policy. - The probabilities that the student $A$ and $B$ pass an examination are $\dfrac{2}{3}$
and $\dfrac{3}{4}$ respectively. Find the probabilities that:
(a) both $A$ and $B$ pass the examination.
(b) exactly one of $A$ and $B$ passes the examination. - Three groups of children consists of $3$ boys and $1$ girl, $2$ boys and $2$ girls, and $1$ boy and $3$ girls respectively. If a child is chosen from each group, find the probability that $1$ boy and $2$ girls are chosen.
%206aa.png)
$\text{ Let } B = \text{ boy, } R = \text{ red, } G = \text{ girl }$Exercise $7.3$
- After a large number of tossing a pin, the probability of 'pin up' was estimated to be $0.3$. In $400$ more trials, how many times would 'pin up' be expected
- If a die is rolled $60$ times, what is the expected frequency of
(a) $1 \text{ turns up? }$ (b) $\text{ a number divisible by } 3 \text{ turns up? }$ (c) $\text{ a factor of } 6 \text{ turns up? }$ - Two honest coins are tossed. How many times would you expect to obtain two heads in $200$ trials?
- The probability of scoring $12$ when throwing two dice at once is $\dfrac{1}{36}$. If such an experiment is repeated $720$ times, what is the expected frequency of the score not being $12$?
- A spinner is equally lilely to point to any one of the numbers: $1$, $2$, $3$, ..., $10$.
(a) What is the probability of an odd number? (b) What is the probability of an even number? (c) If the arrow is spun $1000$ times, what final score would you expect if all the individual scores are added together?
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