Question

. A bag contains 6 red and 3 black chips. One chip is selected, its color is recorded, and it is returned to the bag. This process is repeated until 5 chips have been selected. What is the probability that one red chip was selected?

Answer 1

0.0412

Explanation:

Total chips = 6 red + 3 black chips

Total chips=9

n=5

Probability of (Red chips ) can be determined by

=
`(6)/(9)`

=
`(2)/(3)`

=0.667

Now we used the binomial theorem

`P(x) = C(n,x)*px*(1-p)(n-x).....Eq(1)\\ putting \ the \ given\ value \ in\ Eq(1)\ we \ get \\p(x=1) = C(5,1) * 0.667^1 * (1-0.667)^4`

This can give 0.0412

Answer 2

The probability that one red chip was selected is 0.0053.

Explanation:

Let the random variable X be defined as the number of red chips selected.

It is provided that the selections of the n = 5 chips are done with replacement.

This implies that the probability of selecting a red chip remains same for each trial, i.e. p = 6/9 = 2/3.

The color of the chip selected at nth draw is independent of the other selections.

The random variable X thus follows a binomial distribution with parameters n = 5 and p = 2/3.

The probability mass function of X is:

`P(X=x)={5\choose x}\ ((2)/(3))^(x)\ (1-(2)/(3))^(5-x);\ x=0,1,2...`

Compute the probability that one red chip was selected as follows:

`P(X=1)={5\choose 1}\ ((2)/(3))^(1)\ (1-(2)/(3))^(5-1)`

`=5*(2)/(3)* (1)/(625)\\\\=\farc{2}{375}\\\\=0.00533\\\\\approx 0.0053`

Thus, the probability that one red chip was selected is 0.0053.