In a random sample of three integers between 1 and 365 inclusive, what
is the probability, to three decimal place accuracy, that
(a) the three
numbers will all be di�fferent;
(b) the three numbers will not all be
di�fferent?
To find the probability in this case, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.
First, let's find the total number of possible outcomes. Since we are selecting three integers from a set of 365 integers, each with equal probability of being chosen, the total number of possible outcomes is given by:
Total possible outcomes = 365 * 365 * 365
Now, let's find the number of favorable outcomes for each scenario.
(a) The three numbers will all be different.
To determine the number of favorable outcomes, we can think of it as selecting three integers one by one. For the first selection, there are 365 possible choices. For the second selection, there are 364 remaining choices (since we want the numbers to be different). For the third selection, there are 363 remaining choices.
Number of favorable outcomes for scenario (a) = 365 * 364 * 363
(b) The three numbers will not all be different.
To determine the number of favorable outcomes for this scenario, we need to subtract the number of favorable outcomes for scenario (a) from the total number of possible outcomes.
Number of favorable outcomes for scenario (b) = Total possible outcomes - Number of favorable outcomes for scenario (a)
Now, we can calculate the probabilities.
(a) Probability for the three numbers being different = Number of favorable outcomes for scenario (a) / Total possible outcomes
(b) Probability for the three numbers not all being different = Number of favorable outcomes for scenario (b) / Total possible outcomes
Let's now calculate the probabilities to three decimal place accuracy.
Total possible outcomes = 365 * 365 * 365 = 48,627,125
Number of favorable outcomes for scenario (a) = 365 * 364 * 363 = 47,863,380
Number of favorable outcomes for scenario (b) = Total possible outcomes - Number of favorable outcomes for scenario (a) = 48,627,125 - 47,863,380 = 763,745
(a) Probability for the three numbers being different = 47,863,380 / 48,627,125 ≈ 0.983
(b) Probability for the three numbers not all being different = 763,745 / 48,627,125 ≈ 0.016