Question

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is:

• A. 0.3024
• B. 0.4235
• C. 0.6976
• D. 0.8125

Hint:

When asked for "at least one match", always compute "no matches" first and subtract from 1.

Answer 1

The Correct Option is C

We want the probability that, when five chocolates are drawn (one from each identical bag), at least two of them are the same.

Step 1: Use complement probability.
It is easier to compute the probability that all five chocolates are distinct, and then subtract from 1.

Step 2: Calculate probability that all five picks are different.
Each bag contains the same 10 distinct chocolates.
The first pick can be any chocolate: probability = \(1\).
The second pick must be different from the first: probability = \(\frac{9}{10}\).
The third pick must be different from the first two: \(\frac{8}{10}\).
The fourth pick must be different from the first three: \(\frac{7}{10}\).
The fifth pick must be different from the first four: \(\frac{6}{10}\).
Thus, \[ P(\text{all distinct}) = 1 \cdot \frac{9}{10} \cdot \frac{8}{10} \cdot \frac{7}{10} \cdot \frac{6}{10} = 0.3024. \]

Step 3: Use complement rule.
\[ P(\text{at least two identical}) = 1 - P(\text{all distinct}) = 1 - 0.3024 = 0.6976. \]

Step 4: Conclusion.
Thus, the probability that at least two chocolates match is \(0.6976\).