Question

A bag contains five white balls and five black balls. Your goal is to draw two black balls.

Answer 1

Answer:

a.) the probability that both the ball drawn are black  = $\frac{10}{45} = \frac{2}{9}$

b.) Sum of all the probabilities = $\frac{\frac{2}{9} }{1 - \frac{2}{9} } = \frac{\frac{2}{9} }{\frac{8}{9} } = \frac{2}{8} =\frac{1}{4} = 0.25$

Step-by-step explanation:

a) two balls are drawn at random .

   total number of balls   =  10

 number of black balls    =  5

  number of white balls    = 5.

   the number of ways two balls can be drawn = $\binom{10}{2} = \frac{10!}{2! 8!} = \frac{10\times9}{2} = 45$

  the number of ways two black balls can be drawn = $\binom{5}{2} = \frac{5!}{2! 3!} = \frac{5\times4}{2} = 10$

   the probability that both the ball drawn are black  = $\frac{10}{45} = \frac{2}{9}$

b) probability that First draw of two black balls = $\frac{2}{9}$

  probability that first draw is of two white balls is and second draw is of

  two black balls  = $\frac{2}{9} \times\frac{2}{9}$

  Probability that first two draws are of white balls and the third draw is of

  two black balls = $\frac{2}{9} \times\frac{2}{9} \times\frac{2}{9}$

  This is a geometric sequence and the final probability will be the sum of    

  all such probabilities, where we can take the the sequence to be an infinite series and the first term is $\frac{2}{9}$ and the common ratio is $\frac{2}{9}$  which is less than 1.

  Sum of all the probabilities = $\frac{\frac{2}{9} }{1 - \frac{2}{9} } = \frac{\frac{2}{9} }{\frac{8}{9} } = \frac{2}{8} =\frac{1}{4} = 0.25$