Hi everyone!

I need a little help in answering part (iii) to this Question.

Thanks!!

(edit: it won't let me attach the file so i'm going to type out the Q below)

*One bag contains 5 blue and 3 yellow balls and another bag contains 7 blue and 2 yellow balls. *

iii) A bag is chosen at random and a ball drawn out. The result is recorded, then the ball is placed back in the bag. If this is done 20 times, find the probability of drawing out 12 blue balls correct to 3 decimal places.

Hey annabeljxde,

Let's start off by supposing we choose the bag with 5 blue and 3 yellow balls. Let's call this Bag A. We have a 1/2 chance of choosing Bag A, as the choice of each bag is equally likely.

Now, since the balls are replaced, the probabiliy of choosing a blue ball in each selection is 5/8 (blue balls / total balls).

We now establish a binomial expression for our 20 selections. This will be:

(P(choosing blue) + P(not choosing blue))

^{number of selections} = (5/8 +3/8)

^{20}Using the binomial theorem, the probability of choosing 12 blue balls in 20 selections is: 20C12 x (5/8)

^{12} x (3/8)

^{8}Remembering that we had a 50% chance of choosing Bag A, we can say that the probability of choosing 12 blue balls AFTER choosing Bag A is:

1/2 x 20C12 x (5/8)

^{12} x (3/8)

^{8}However, this is only half our solution. We repeat exactly the same thing using the bag with 7 blue and 2 yellow balls (we'll call this Bag B).

Again, we have a 1/2 probability of choosing Bag B. We have a 7/9 chance of choosing a blue ball in each turn we take.

Establishing a binomial expression for 20 selections, we obtain (7/9 +2/9)

^{20}Now, the probability of choosing 12 blue balls becomes 20C12 x (7/9)

^{12} x (2/9)

^{8}Remembering that we had a 50% chance of choosing Bag B in the first place, the probability of choosing 12 blue balls from bag B is:

1/2 x 20C12 x (7/9)

^{12} x (2/9)

^{8}For the total probability, we just add these two separate probabilities (since we can choose either bag A OR bag B to start with - when you see OR, it generally means addition).

And so, we wind up with:

P(choosing 12 blue) = 1/2 x 20C12 x (5/8)

^{12} x (3/8)

^{8} + 1/2 x 20C12 x (7/9)

^{12} x (2/9)

^{8} = 0.106 (3 d.p).

Hopefully this is all correct.