Suppose you have 3 bags containing only apples and oranges. bag a has 2 apples and 4 oranges, bag b has 8 apples and 4 oranges, and bag c has 1 apple and 3 oranges. you pick 1 fruit (at random) from each bag. a) what is the probability that you picked exactly 2 apples? b) suppose you picked 2 apples but forgot which bag they came from. what is the probability that you picked an apple from bag a?

Step-by-step explanation:

There are three bags

Bag A

2apples and 4 oranges

P(A¹)=2/6

P(A¹)=⅓

P(O¹)=4/6

P(O¹)=⅔

Bag B

8 apples and 4 oranges

P(A²)=8/12

P(A²)=⅔

P(O²)=4/12

P(O²)=⅓

Bag C

1 apple and 3 oranges

P(A³)=¼

P(O³)=¾

Note

P(A¹) means probability of Apple in bag A

P(A²) means probability of Apple in bag B

P(A³) means probability of Apple in bag C

P(O¹) means probability of oranges in bag A.

P(O²) means probability of oranges in bag B.

P(O³) means probability of oranges in bag C.

a. The probability of picking exactly two apples can be analyzed as

Picking apple in bag A and picking apple in bag B and picking orange in bag C or picking apple in bag A and picking orange in bag B and picking apple in bag C or picking orange in bag A and picking apple in bag B and picking apple in bag C.

Then,

P(exactly two apples)=P(A¹ n A² n O³) + P(A¹ n O² n A³) + P(O¹ n A² n A³)

Since they are mutually exclusive

Then,

P(exactly two apples) =

P(A¹) P(A²)P(O³) + P(A¹)P(O²)P(A³) + P(O¹) P(A²) P(A³)

P(exactly two apples) =

(⅓×⅔×¾)+(⅓×⅓×¼)+(⅔×⅔¼)

P(exactly two apples)=1/6 +1/36 +1/9

P(exactly two apples)= 11/36

b. Probability that an apple comes from Bag A out of the two apple will be Picking apple in bag A and picking apple in bag B and picking orange in bag C or picking apple in bag A and picking orange in bag B and picking apple in bag C.

P(an apple belongs to bag A)=P(A¹ n A² n O³) + P(A¹ n O² n A³)

Since they are mutually exclusive

Then,

P(an apple belongs to bag A) =

P(A¹) P(A²)P(O³) + P(A¹)P(O²)P(A³)

P(an apple belongs to bag A) =

(⅓×⅔×¾)+(⅓×⅓×¼)

P(an apple belongs to bag A)=1/6 +1/36

P(an apple belongs to bag A)= 7/36

the answer is ! ?