For this problem, assume the balls in the box
are numbered 1 through 8, and that an experiment consists of randomly
selecting 3 balls one after another without replacement.
What probability should be assigned to the event that at least one ball has
an odd number?
To calculate the probability of at least one ball having an odd number, we need to first determine the total number of possible outcomes and the number of outcomes that satisfy the event.
Total number of outcomes:
Since we are selecting 3 balls one after another without replacement, we can find the total number of outcomes using the concept of permutations. We have 8 balls to choose from for the first selection, 7 balls for the second selection, and 6 balls for the third selection. Therefore, the total number of outcomes is:
Total Outcomes = 8 * 7 * 6 = 336
Number of outcomes satisfying the event:
To find the number of outcomes where at least one ball has an odd number, we can consider the different scenarios. There are 4 odd-numbered balls (1, 3, 5, 7) and 4 even-numbered balls (2, 4, 6, 8).
Scenario 1: One odd-numbered ball and two even-numbered balls
- The odd-numbered ball can be chosen in 4 ways
- The first even-numbered ball can be chosen in 4 ways (since there are 4 even-numbered balls to choose from)
- The second even-numbered ball can be chosen in 3 ways (since there are only 3 even-numbered balls left to choose from)
- Therefore, the number of outcomes in Scenario 1 is: 4 * 4 * 3 = 48
Scenario 2: Two odd-numbered balls and one even-numbered ball
- The first odd-numbered ball can be chosen in 4 ways
- The second odd-numbered ball can be chosen in 3 ways (since there are 3 odd-numbered balls left to choose from)
- The even-numbered ball can be chosen in 4 ways
- Therefore, the number of outcomes in Scenario 2 is: 4 * 3 * 4 = 48
Scenario 3: Three odd-numbered balls
- The first odd-numbered ball can be chosen in 4 ways
- The second odd-numbered ball can be chosen in 3 ways
- The third odd-numbered ball can be chosen in 2 ways (since there are only 2 odd-numbered balls left to choose from)
- Therefore, the number of outcomes in Scenario 3 is: 4 * 3 * 2 = 24
Total number of outcomes satisfying the event:
To get the total number of outcomes satisfying the event, we sum up the number of outcomes from each scenario:
Total Outcomes Satisfying the Event = 48 + 48 + 24 = 120
Probability:
The probability of an event is calculated by dividing the number of outcomes satisfying the event by the total number of possible outcomes. Therefore, the probability of at least one ball having an odd number is:
Probability = Total Outcomes Satisfying the Event / Total Outcomes
Probability = 120 / 336
Probability = 0.3571 (rounded to four decimal places)
So, the probability assigned to the event that at least one ball has an odd number is approximately 0.3571.