Question

The numbers​ 1, 2,​ 3, 4, and 5 are written on slips of​ paper, and 2 slips are drawn at random one at a time without replacement. ​(a) Find the probability that the first number is 4​, given that the sum is 9. ​(b) Find the probability that the first number is 3​, given that the sum is 8.

Answer 1

Answer:

(a)

The probability is :  1/2

(b)

The probability is :  1/2

Step-by-step explanation:

The numbers​ 1, 2,​ 3, 4, and 5 are written on slips of​ paper, and 2 slips are drawn at random one at a time without replacement.

The total combinations that are possible are:

(1,2)   (1,3)    (1,4)    (1,5)

(2,1)   (2,3)   (2,4)   (2,5)

(3,1)   (3,2)   (3,4)   (3,5)

(4,1)   (4,2)   (4,3)   (4,5)

(5,1)   (5,2)   (5,3)   (5,4)

i.e. the total outcomes are : 20

(a)

Let A denote the event that the first number is 4.

and B denote the event that the sum is: 9.

Let P denote the probability of an event.

We are asked to find:

               P(A|B)

We know that it could be calculated by using the formula:

$P(A|B)=\dfrac{P(A\bigcap B)}{P(B)}$

Hence, based on the data we have:

$P(A\bigcap B)=\dfrac{1}{20}$

( Since, out of a total of 20 outcomes there is just one outcome which comes in A∩B and it is:  (4,5) )

and

$P(B)=\dfrac{2}{20}$

( since, there are just two outcomes such that the sum is: 9

(4,5) and (5,4) )

Hence, we have:

$P(A|B)=\dfrac{\dfrac{1}{20}}{\dfrac{2}{20}}\\\\i.e.\\\\P(A|B)=\dfrac{1}{2}$

(b)

Let A denote the event that the first number is 3.

and B denote the event that the sum is: 8.

Let P denote the probability of an event.

We are asked to find:

               P(A|B)

Hence, based on the data we have:

$P(A\bigcap B)=\dfrac{1}{20}$

( since, the only outcome out of 20 outcomes is:  (3,5) )

and

$P(B)=\dfrac{2}{20}$

( since, there are just two outcomes such that the sum is: 8

(3,5) and (5,3) )

Hence, we have:

$P(A|B)=\dfrac{\dfrac{1}{20}}{\dfrac{2}{20}}\\\\i.e.\\\\P(A|B)=\dfrac{1}{2}$

Answer 2

Using the probability concept, it is found that:

a) 0.5 = 50% probability that the first number is 4​, given that the sum is 9.

b) 0.5 = 50% probability that the first number is 3​, given that the sum is 8.

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• A probability is the number of desired outcomes divided by the number of total outcomes.

The possible outcomes are:

(1,2), (1,3), (1,4), (1,5) .

(2,1), (2,3), (2,4), (2,5).

(3,1), (3,2), (3,4), (3,5).

(4,1), (4,2), (4,3), (4,5).

(5,1), (5,2), (5,3), (5,4).

Item a:

• There are 2 outcomes with a sum of 9, which are (4,5) and (5,4).
• On one of them, (5,4), the first term is 4.

Then:

$p = \frac{1}{2} = 0.5$

0.5 = 50% probability that the first number is 4​, given that the sum is 9.

Item b:

• There are 2 outcomes with a sum of 8, (3,5) and (5,3).
• On one of them, (3,5), the first term is 5.

Then:

$p = \frac{1}{2} = 0.5$

0.5 = 50% probability that the first number is 3​, given that the sum is 8.

A similar problem is given at https://brainly.com/question/24262336