In a shop there are 20 customers, 18 of whom will make a purchase. If three customers are selected, one at a time, at random, what is the probability that all will make a purchase?
For the probability of all events, multiply the individual probabilities.
Without replacement, 18/20 * 17/19 * 16/18 = ?
To find the probability that all three selected customers will make a purchase, we need to consider the total number of possible outcomes and the number of favorable outcomes.
Let's break it down step by step:
Step 1: Determine the total number of possible outcomes.
Since we are selecting three customers from a total of 20, and order does not matter, we can use combinations. The number of combinations of selecting three customers out of 20 is given by the binomial coefficient "20 choose 3" or C(20, 3), which can be calculated using the formula: C(n, r) = n! / (r!(n - r)!).
C(20, 3) = 20! / (3!(20 - 3)!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140
So, there are 1140 possible outcomes when selecting three customers from a total of 20.
Step 2: Determine the number of favorable outcomes.
Out of the 20 customers, 18 are expected to make a purchase. We want all three selected customers to be among those 18 who will make a purchase. Since the order does not matter, we can again use combinations.
The number of combinations of selecting three customers out of 18 is given by the binomial coefficient "18 choose 3" or C(18, 3).
C(18, 3) = 18! / (3!(18 - 3)!) = (18 * 17 * 16) / (3 * 2 * 1) = 816
So, there are 816 favorable outcomes where all three customers will make a purchase.
Step 3: Calculate the probability.
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability = Number of favorable outcomes / Number of possible outcomes
= 816 / 1140
= 0.716
Thus, the probability that all three selected customers will make a purchase is approximately 0.716 or 71.6%.