Question

involving the selection of two apples from a bag of red and yellow apples without replacement. Assume that the bag has a total of 14 apples: 10 red and 4 yellow. (1) What is the probability that the second apple you pick is red? equation editorEquation Editor (2) What is the probability that you pick at least one red apple?

Answer 1

(1)
`(5)/(7)` is the probability that the second apple you pick is red

(2)
`(85)/(91)` is the probability that you pick at least one red apple

Explanation:

(1)

First you must remember that the probability is:

`(desired outcome)/(total possible outcomes)`

There are 2 ways this can happen independently: either the first apple and our second apple are both red. Or the first apple is yellow, and the second is red.

You know you have 14 apples in total: 10 red and 4 yellow.

In the first case, considering the definition, the probability that the first apple is red is:
`(10)/(14)`

This means that of the 14 apples in total, you have 10 that are red.

Now that you picked an apple and it was red, there were 13 apples in total in the bag, and 9 red apples. So the probability that the second apple you pick is red is
`(9)/(13)`.

So, to get the probability that both apples are red, you get them by multiplying, because the probability of the second selection depends on the first:

`(10)/(14) *(9)/(13) =(45)/(91)`

Now consider the case where the first apple is yellow and the second is red. For that, considering the definition, the probability that the first apple is yellow is:
`(4)/(14)`

This means that of the 14 apples in total, you have 4 that are yellow.

Now that you picked an apple and it was yellow, there were 13 apples in total in the bag. But the red apples are still 10, because the apple you picked was yellow. So the probability that the second apple you pick is red is
`(10)/(13)`.

So the probability that only the second apple is red is:

`(4)/(14) *(10)/(13)=(20)/(91)`

To find the chance that one or the other happens, you add:

`(45)/(91)+(20)/(91)=(65)/(91)=(5)/(7)`

So,
`(5)/(7)` is the probability that the second apple you pick is red

2)

The probability that in 2 selections at least one is red is: 1 – probability of both yellow, calculating the probability that both are yellow following the previous reasoning :

`1-(4)/(14) *(3)/(13) =1-(6)/(91)=(85)/(91)`

So,
`(85)/(91)` is the probability that you pick at least one red apple

Answer 2

1) The probability that the second apple is red is 0.7143 (71.43%).

2) The probability that at least one red apple is picked 0.9341 (93.41%).

Explanation:

We can make a probability tree as the attached picture.

1) There are 2 ways in the probability tree when the second apple is red:

Both apples are red:

P(R∩R)=
`(10)/(14)(9)/(13)`=0.4945

Only the second apple is red:

P(Y∩R)=
`(4)/(14)(10)/(13)`=0.2198

The probability that the second apple is red is the sum of the previous probabilities.

P(2nd R)=P(R∩R)+P(Y∩R)=0.4945+0.2198=0.7143

2) To find the probability that at least one apple is red, we can get the probability of none of the apples is red and then it will be subtracted from 1.

The way in the probability tree is Y∩Y:

P(Y∩Y)=
`(4)/(14) (3)/(13)`=0.0659

P(at least 1 R)= 1-P(Y∩Y)=1-0.0659=0.9341