Probability !
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)=

A

Step-by-step 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!

its a straight line parallel to the time axis