A bag contains five white b***s and three black b***s. Your goal is to draw two black b***s.

a) You simultaneously draw two b***s at random. What is the probability that they are both black?

b) You simultaneously draw two b***s at random. Once you have drawn two b***s, you put back any white b***s, and redraw so that you again have two drawn b***s. What is the probability that you now have two black b***s? (Include the probability that you choose two black b***s on the first draw.)

The answer for b is **not** 45/56, thanks!

Keihaku Nov 3, 2023

#1**0 **

a)

There are 8 ways to choose 2 balls out of 8 total.

There are 3 ways to choose 2 black balls out of 3 black.

The probability is 3/8.

b)

The probability of drawing two black balls on the first draw is 3/8, as you already calculated.

If you don't draw two black balls on the first draw, then you must draw at least one white ball. When you put the white ball back and draw again, you have a 82 chance of drawing another white ball, and a 83 chance of drawing a black ball.

Therefore, the probability of drawing two black balls, including the possibility of drawing two black balls on the first draw, is:

(3/8) + (5/8) * (3/8) = 9/16

parmen Nov 5, 2023

#3**0 **

Start with 5 white and 3 black

1) P(BB) \(= \frac{3}{8}*\frac{2}{7}=\frac{6}{56}\)

P(BW) = \(\frac{3}{8}*\frac{5}{7}=\frac{15}{56}\)

throw back the white ball then P(drawing black) \(=\frac{2}{7}\)

2) So Prob of getting BWB \(\frac{15}{56}*\frac{2}{7}= \frac{15}{196}\)

3) The prob of getting WBB is the same as BWB

P(WW) \(\frac{5}{8}*\frac{4}{7}=\frac{20}{56}\)

Throw both balls back and the prob of then getting BB is 6/56

4) P( WWBB) = (20/56)*(6/56) = \frac{15}{392}

So the prob of ending up with 2 black balls is \(\frac{15}{56}+\frac{15}{196}+\frac{15}{196}+\frac{15}{392}= \frac{45}{98}\)

Melody Nov 9, 2023

#5**0 **

Hi Melody, thank you for helping me! I don't quite understand the formatting of your answer so if you could help me understand which part is the answer to a) and b) that would be great :)

p.s. I entered 45/98 for part b) and it said incorrect, which is the reason why I'm asking for help, in case I'm the one who misread smth. I just want to understand what the answers are and how you got them so I can apply the formula to other problems I have :) Thanks!

Keihaku
Nov 10, 2023