Question

You are once again at the ice cream shop, but you are feel like eating healthy. There are 6 ice creams and 5 frozen yogurts that look very similar. 3 of the ice creams are low fat and 2 of the yogurts are low fat. You scoop 1 scoop each from 2 different barrels. What is the possibility that both of the scoops are low fat, and 1 is ice cream and the other is yogurt?

Answer 1

Ok, let's assume that the first scoop is low-fat ice cream.

The probability for this event is equal to the quotient between the number of barrels of low-fat ice cream (we know that there are 3) and the total number of barrels.

Assuming that both, ice-cream and frozen yogurt, come in barrels, then the total number of barrels is 6 + 5 = 11

there are 11 total barrels.

Then the probability of getting frozen ice cream in the first scoop is:

p = 3/11

Now in the second scoop, we will get the low-fat frozen yogurt.

The probability is computed in the same way, the quotient between the number of low fat frozen yogurts (2) and the total number of barrels (10 in this case, because the scoops are from different barrels)

so now the probability is

q = 2/10

The joint probability is the product between the individual probabilities, this gives:

P = (3/11)*(2/10) = 6/110

And there is a single permutation, the case where first we get the frozen yogurt and after that the ice-cream, so there is a permutation of 2, so we need to add a factor of 2:

Probability = 2*P = 2*(6/110) = 12/110