There are 4 baskets. The first basket has 3 apples and 4 oranges, the second one has 4 apples and 5 mangoes, the third one has 6 Mangoes and 2 bananas and the last one has 7 bananas and 2 apples. If a fruit is randomly chosen from any basket and it comes out to be an apple, then what is the probability that it was taken out from the second basket?
Have you done Baye's theorem, total probability and conditional probability?
If not, you need to review these topics before you can tackle this problem.
Here the assumption is not clear, but I will assume that it is equally probable to
1. pick one of the 4 baskets
2. pick any fruit of a given basket.
Using notation of condition probability:
P(A|B)= probability of event A given B has happened.
The baskets and the corresponding events will be named 1-4.
Event of choosing an apple is A.
Then
P(A|1)=3/7
P(A|2)=4/9
P(A|3)=0/8
P(A|4)=2/9
Hence event 1,2,3,4 are the partitions of the sample space.
The probability of choosing an apple, by the law of total probability
P(A)=P(A|1)P(1)+P(A|2)P(2)+P(A|3)P(3)+P(A|4)P(4)
=3/7*1/4+4/9*1/4+0/8*1/4+2/9*1/4
=23/84
By Baye's theorem,
P(2|A)=P(A|2)*P(2)/P(A)
=4/9*(1/4)/(23/84)
=28/69
Probability of Picking an apple from any one of the boxes should be more than picking an apple from a specific box.
But your application of Baye's Theorem for this question gives the probability the other way. can you explain that
Well, this question seems a-peeling! To find the probability, we need to calculate the total number of apples and then consider the number of apples in the second basket.
The first basket has 3 apples, the second has 4 apples, the third has 2 apples, and the last basket also has 2 apples. So, in total, we have 3 + 4 + 2 + 2 = 11 apples.
Out of these 11 apples, 4 of them are in the second basket. Therefore, the probability of choosing an apple from the second basket is 4/11.
So, it looks like the probability is fruitfully 4/11!
To find the probability that the apple was taken out from the second basket, we need to calculate the probability of picking an apple from the second basket and divide it by the overall probability of picking any apple from any basket.
Step 1: Calculate the probability of picking an apple from the second basket.
In the second basket, there are a total of 4 apples and 5 mangoes. So, the probability of picking an apple from the second basket is 4/(4+5) = 4/9.
Step 2: Calculate the overall probability of picking any apple from any basket.
To do this, we need to find the total number of apples in all the baskets. Summing the number of apples in each basket, we get: 3+4+2 = 9.
So, the overall probability of picking any apple from any basket is 9/(3+4+4+5+6+2+7+2) = 9/33 = 3/11.
Step 3: Divide the probability of picking an apple from the second basket (Step 1) by the overall probability of picking any apple from any basket (Step 2).
Therefore, the probability that the apple was taken out from the second basket is (4/9) / (3/11) = (4/9) * (11/3) = 44/27.
The final probability is 44/27.