Without consultation, each of five organizations announces a one-day convention to be held during July. Find the probability that at least two organizations specify the same day for their convention.
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To find the probability that at least two organizations specify the same day for their convention, we can use the principle of inclusion-exclusion.
Let's break it down step by step:
Step 1: Calculate the total number of ways each organization can choose a day for their convention. Since there are 31 days in July, each organization has 31 choices. Therefore, the total number of ways is 31^5.
Step 2: Calculate the number of ways that no organizations specify the same day for their convention. The first organization has 31 choices for their convention day, the second organization has 30 choices (excluding the day chosen by the first organization), the third has 29 choices, and so on. Therefore, the number of ways that no organizations specify the same day is 31 * 30 * 29 * 28 * 27.
Step 3: Calculate the number of ways that exactly one organization specifies the same day for their convention. There are 5 organizations, and each organization has 31 choices for their convention day. So, there are 5 * 31 choices for the organization that specifies the same day, and the other 4 organizations can choose from the remaining 30 days. Therefore, the number of ways that exactly one organization specifies the same day is 5 * 31 * 30 * 29 * 28.
Step 4: Calculate the probability that at least two organizations specify the same day. We can use the principle of inclusion-exclusion here. The probability is equal to the number of ways that at least two organizations specify the same day divided by the total number of ways.
So, the probability = (Number of ways that at least two organizations specify the same day) / (Total number of ways)
Number of ways that at least two organizations specify the same day = Total number of ways - (Number of ways that no organizations specify the same day + Number of ways that exactly one organization specifies the same day)
Substituting the values we calculated earlier:
Number of ways that at least two organizations specify the same day = 31^5 - (31 * 30 * 29 * 28 * 27 + 5 * 31 * 30 * 29 * 28)
Total number of ways = 31^5
Now you can calculate the probability by dividing the number of ways by the total number of ways.