Question

From a bowl containing five red, three white, and seven blue chips, select four at random and without replacement. Compute the conditional probability of one red, zero white, and three blue chips, given that there are at least three blue chips in this sample of four chips.

Short answer

The conditional probability of one red, zero white, and three blue chips, given that there are at least three blue chips in this sample of four chips can be solved by using the combination formula and conditional probability formula and the answer is \(_{5}C_{1}\) * \(_{7}C_{3}\) / (_{5}C_{1} * _{7}C_{3} + _{7}C_{4}).

Step-by-step solution

Calculate total number of ways to select four chips

The total number of ways to select four chips without replacement from the fifteen (5 red + 3 white + 7 blue) is given by \(_{15}C_{4}\), using the combination formula, where n is the total number of items and r is the number of items to select.

Calculate the number of ways to select three blue and one red chip

The number of ways to select one red chip out of five is \(_{5}C_{1}\). Likewise, the number of ways to select three blue chips out of seven is \(_{7}C_{3}\). The total number of ways to select one red and three blue chips is the product of these two, i.e. \(_{5}C_{1}\) * \(_{7}C_{3}\).

Calculate the conditional probability

According to the conditional probability formula, the probability of the event (selecting one red, zero white, and three blue chips) given another event (there are at least three blue chips) is the probability of the intersection of these two events occurring divided by the probability of the latter event occurring. The probability of selecting at least three blue chips can be calculated as the sum of the probability of selecting three blue chips and one chip of any other color and the probability of selecting four blue chips. Thus, the conditional probability will be \(_{5}C_{1}\) * \(_{7}C_{3}\) / (_{5}C_{1} * _{7}C_{3} + _{7}C_{4}).