Probability Question

Question

A bag contains 5 white balls and 8 red balls. Three balls are drawn, without replacement, from the bag. (Enter your probabilities as fractions.)

(a) What is the probability that all three balls are white? 

 

(b) What is the probability that exactly one ball is white? 

 

(c) What is the probability that at least one ball is white? 

Answer 1

(a) 5/13 * 4/12* 3/11

(b) (5 choose 1) / (13 choose 5)

(c) {(5 choose 1) + (5 choose 2) + (5 choose 3)} / (13 choose 3)

Answer 2

All right, here we go!

So we have:

5 white balls

8 red balls

13 total balls

a) Probability that all three balls are white = P(WWW)

The probability of one ball being white is = # of white balls

total # of balls

WithOUT replacement means we do not put the white ball back, we place it to the side. So that ball can no longer be picked again. So after we pick one white ball, how many balls are left? Still 13? But we placed one to the side. That's right, there are 12 left. And are there still 5 white balls? Remember we placed one white to the side.

so P(of thef irst one being white) is = 5/13

The P (of the second one being white) is = 4/12 (why 4 and 12? Think about where that first white ball is)

The P (of the third ball being white) is = 3/11

If we want to find the P (WWW) we multiply all these fractions together.

So, P(WWW) = 5 x 4 x 3 = 60 *(x is multiply)

13 12 11 1716

Can you reduce this fraction? First start by dividing both the top and bottom by 2. Keep simplifying from there!

b) Probability that exactly and ONLY one ball is white. It doesn't matter if the first ball we choose is white or the the last ball, so we are looking for P(WRR) aka white and then non white.

Find the probability of each one and let's multiply them together.

P (WRR) = 5 x 8 x 7

13 12 11

Question? We start off with 8 red balls, because that's how many are in the bag. Why does it go down to 7? And why does it go 13, 12, 11?

Answer: We are doing this withOUT replacing the ball back in the bag. Imagine you are holding that marble in your hand. So after you hold the first red ball, there are no longer 8 red balls. Only 7!

Multiply the above: P(WRR) = 280 Can you reduce? Simplify!

1716

c) Probablity that AT LEAST one ball is white. So we could have ONE white ball, or TWO white balls, or even THREE white balls. We just DON'T want all red balls. It might be easier to find the probability of ALL red balls.

Remember: The Probabilility of ALL red + the Probability of ONE being white (aka not all red) = 1

P (RRR) = 8 x 7 x 6 = 336 Can you reduce? Of course you can! Great job!

13 12 11 1716

I hope that helps! Let me know if you have any questions.

-Margaret

Answer 3

The total combination of picking 3 balls out of 13 balls is ₁₃C₃ = 13x12x11/(1x2x3)

a) The total combination of picking 3 white balls is ₅C₃ = 5x4x3/(1x2x3), making the probability 5/143.

b) The total combination of picking 1 white balls and 2 red balls is ₅C₁ x ₈C₂ = (5x8x7)/(1x2), making the probability 70/143

c) The total combination of no ball is white, which is the same all red balls, is ₈C₃ = (8x7x6)/(1x2x3), making the probability 28/143. This means that the probability of at least one ball is white is 1 -28/143 = 115/143.