Without consultation, each of four 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.
To find the probability that at least two organizations specify the same day for their convention, we can consider the complementary probability.
Step 1: Determine the total number of possible outcomes.
Since each organization can choose any day in July independently, the total number of possible outcomes is 31^4 since there are 31 days in July.
Step 2: Determine the number of favorable outcomes.
The number of favorable outcomes is the number of ways in which at least two organizations can specify the same day for their convention.
To calculate this, we can use the principle of inclusion-exclusion:
Number of ways at least two organizations specify the same day =
(Number of ways two organizations specify the same day) -
(Number of ways three organizations specify the same day) +
(Number of ways four organizations specify the same day)
Step 3: Calculate the number of ways two organizations specify the same day.
There are 4C2 ways to choose 2 organizations out of 4, and for each pair, there is only 1 way for them to specify the same day. Therefore, the number of ways two organizations specify the same day is 4C2 * 1 = 6.
Step 4: Calculate the number of ways three organizations specify the same day.
Similarly, there are 4C3 ways to choose 3 organizations out of 4, and for each triple, there are 31 possible days they can choose, except for the case where all three organizations specify the same day, which is already counted in Step 3. Therefore, the number of ways three organizations specify the same day is (4C3 * 31) - 6 = 118.
Step 5: Calculate the number of ways four organizations specify the same day.
There is only 1 day in July that all four organizations can choose, so the number of ways four organizations specify the same day is 1.
Step 6: Calculate the total number of favorable outcomes.
The total number of favorable outcomes is 6 - 118 + 1 = -111.
Step 7: Calculate the probability.
The probability of at least two organizations specifying the same day for their convention is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = number of favorable outcomes / total number of possible outcomes
= -111 / (31^4)
However, it seems that there is an error in the calculations, as the number of favorable outcomes is negative, which is not possible. This indicates that there is no way for at least two organizations to specify the same day for their convention among the given conditions.
To find the probability that at least two organizations specify the same day for their convention, we can use the concept of the complement rule.
First, let's find the total number of ways that the organizations can specify the convention days. Each organization has 31 possible days to choose from since July has 31 days. Since there are 4 organizations, the total number of ways they can choose the convention days is 31^4.
Next, let's find the number of ways that no two organizations specify the same day for their convention. In this case, we need to choose different days for each organization.
For the first organization, there are 31 choices for the convention day.
For the second organization, there are 30 choices remaining (one less than the first organization's choice).
For the third organization, there are 29 choices remaining.
For the fourth organization, there are 28 choices remaining.
To find the total number of ways that no two organizations specify the same day, we multiply the number of choices for each organization: 31 * 30 * 29 * 28.
Now, let's find the probability that at least two organizations specify the same day. This is equivalent to 1 minus the probability that no two organizations specify the same day.
Probability = 1 - (number of ways that no two organizations specify the same day / total number of ways the organizations can choose the convention days)
Probability = 1 - (31 * 30 * 29 * 28 / 31^4)
Simplifying this fraction will give you the final probability.