Probability of Customers Choosing the Same Smoothie Flavor at a Stand

Imagine a bustling stand where 8 customers are waiting in line to order fruit smoothies from any of 32 different available flavors. What is the probability that at least two of the customers buy the same flavor? This article demystifies the calculation by using the principles of complementary probability.

Understanding the Problem Using Complementary Probability

To find the probability that at least two of the 8 customers buy the same flavor, we first compute the likelihood that all 8 customers choose different flavors. We then subtract this value from 1 to find the probability of at least two customers having the same flavor.

Step-by-Step Calculation

Step 1: Calculate the Total Number of Ways for 8 Customers to Choose Smoothies

Each customer has 32 options. Therefore, the total number of combinations is:

$32^8$

Step 2: Calculate the Number of Ways for 8 Customers to Choose Different Flavors

If all 8 customers choose different flavors, the possibilities are:

$32 times 31 times 30 times 29 times 28 times 27 times 26 times 25$

This can also be expressed using factorial notation:

$frac{32!}{24!}$

Step 3: Calculate the Probability that All Customers Choose Different Flavors

The probability that all customers choose different flavors is:

$P_{all different} frac{32 times 31 times 30 times 29 times 28 times 27 times 26 times 25}{32^8}$

First, calculate the numerator:

$32 times 31 times 30 times 29 times 28 times 27 times 26 times 25 3,711,993,600$

Next, calculate the denominator:

$32^8 10,995,116,277,760$

Thus:

$P_{all different} frac{3,711,993,600}{10,995,116,277,760} approx 0.00338$

Step 4: Calculate the Probability That at Least Two Customers Choose the Same Flavor

The probability that at least two customers choose the same flavor is the complement of the probability that all choose different flavors:

$P_{at least 2 same} 1 - P_{all different} approx 1 - 0.00338 approx 0.99662$

Which can be expressed as a percentage:

$99.66%$

Conclusion

The probability that at least two of the 8 customers buy the same flavor smoothie is approximately 0.99662 or 99.66%. This means it is highly unlikely that all 8 customers will choose different flavors, given the variety and number of customers.