Question

PROBABILITY QUESTION!!!

A jar contains 9 large red marbles, 3 small red marbles, 7 large blue marbles, and 6 small blue marbles. If a marble is chosen at random, which of the conditional probabilities is larger, P(red|small) or P(small|red)

Select one answer and fill in the blank

A) P(red|small) is larger, since P(red|small)= And P(small|red)=

B) P(small|red) is larger, since P(small|red)= And P(red|small)=

C) the probabilities are equal, since P(red|small)=P(small|red)=

Answer 1

A

Explanation:

To solve this you need to use the formulas

P(B|A) = P(A and B)/P(A) and P(A|B) = P(A and B)/P(B)

Let A = all red marbles

Let B = small marbles

Let C = small red marbles

there are 25 total marbles in the bag, then

P(A) = 12/25 (there are 12 red marbles in the bag, big and small)

P(B) = 9/25 (there are 9 small marbles, red and blue)

P(A and B) = 3/25 (there are 3 marbles that are small and red)

To find P(red | small), we use P(A|B) = P(A and B)/P(B) and plug in our known values..

P(red | small) = P( A | B) = (3/25)(9/25)

which simplifies to 3/9, which reduces to 1/3, which equals 0.3333

*Think about it this way too, we know the marble is small, there are 9 small marbles in the bag, 3 are red and 6 are blue, so there is a 3/9 chance of the marble being red if we already know that it's small

To find P(small | red), we use P(B|A) = P(A and B)/P(A) and plug in our known values...

P(small | red) = P(B | A) = (3/25)/(12/25)

which simplifies to 3/12, which reduces to 1/4, which equals 0.25

*Think of it this way too, we know the marble is red, there are 12 red marbles in the bag, 9 are large and 3 are small, so there is a 3/12 chance of the marble being a small red marble if we already know the marble is red

P(red | small) = 0.3333

and

P(small | red) = 0.25

So P(red | small) has the greater probability!