Question

The names of nine boys and six girls from your class are put into a hat. What is the probability that the first two names chosen (one first then the other without replacement)will both be boys' names?

Answer 1

The probability that the first two names selected from a hat of nine boys and six girls will both be boys' names is 12/35. This calculation is done using the hypergeometric distribution, which is appropriate when sampling without replacement from a finite population.

Step-by-step explanation:

The question involves calculating the probability that the first two names drawn from a hat containing the names of nine boys and six girls will be boys' names.

The selection is done without replacement, meaning that once a name is drawn, it is not put back into the hat for subsequent draws.

This scenario can be addressed using the hypergeometric distribution, which is used when you have a finite population and are interested in successes in a sample without replacement.

Here's the step-by-step calculation for the probability:

1. Calculate the probability that the first name drawn is a boy's name:
2. There are 9 boys and a total of 15 students (9 boys + 6 girls), so the probability for the first draw is 9/15 or 3/5.
3. Calculate the probability that the second name drawn is also a boy's name:
4. After one boy's name has been drawn, there are now 8 boys' names left and a total of 14 names in the hat. So the probability for the second draw, given the first was a boy's name, is 8/14 or 4/7.
5. To find the overall probability, multiply the probabilities of the individual events: (3/5) × (4/7).
6. The final probability is therefore: (3/5) × (4/7) = 12/35.

So, the probability that the first two names drawn will both be boys' names is 12/35.