How many riders would there have to be on a bus to yield (a) a 50 percent probability that at least two will have the same birthday? (b) A 75 percent probability? Hint: Use LearningStats.
Please type your Subject in the School Subject: box.
To calculate the number of riders needed on a bus to yield a certain probability that at least two of them will have the same birthday, we can use the concept of the Birthday Paradox. The Birthday Paradox states that in a group of people, the probability of two or more people having the same birthday is higher than what we intuitively expect.
To solve this problem, we can use LearningStats, a statistical formula that helps us calculate the probability of an event.
(a) To find the number of riders needed on a bus to have a 50% probability of at least two riders having the same birthday, we need to calculate the smallest value of n in the equation:
1 - P(n) = 0.5
Where P(n) is the probability that no two riders share the same birthday with n riders.
We can calculate P(n) using the formula:
P(n) = (365/365) * (364/365) * (363/365) * ... * (365 - n + 1)/365
Simplifying the equation:
1 - P(n) = 0.5
P(n) = 0.5
(365/365) * (364/365) * (363/365) * ... * (365 - n + 1)/365 = 0.5
By trying different values of n, you can find that when n = 23, the probability P(n) is approximately 0.492703. Therefore, for a 50% probability, there need to be at least 23 riders on the bus.
(b) To find the number of riders needed on a bus to have a 75% probability of at least two riders having the same birthday, we need to calculate the smallest value of n in the equation:
1 - P(n) = 0.75
Using the same formula as above, trial and error will show that when n = 41, the probability P(n) is approximately 0.736, which is close to 0.75. Therefore, for a 75% probability, there need to be at least 41 riders on the bus.
In summary, for a 50% probability of at least two riders having the same birthday, there need to be 23 riders on the bus, and for a 75% probability, there need to be 41 riders on the bus.