Approximately 80,000 marriages took place in the state of...

Approximately 80,000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couples,
(a) both partners were born on April 30 ;
(b) both partners celebrated their birthday on the same day of the year. State your assumptions.

Key Concepts

Uniform Distribution of Birthdays

This concept assumes that birthdays are equally likely to fall on any day of the year (typically 365 days, ignoring leap years). It forms the basis for calculating the probability of events involving specific birthdays by giving each day an equal chance, which simplifies the probability computations.

Independence of Birthdays

Assuming that the birthdays of individuals are independent means that one person’s birthday does not affect another’s. This independence is crucial when calculating the joint probability for a couple, as it allows the multiplication of individual probabilities to find the likelihood of both partners sharing a specific birthday or the same birthday.

Complement Rule in Probability

The complement rule involves finding the probability of at least one event occurring by subtracting the probability of no events occurring from one. In these kinds of problems, it is often easier to calculate the probability that none of the couples have the given birthday scenario and then subtract that result from 1 to obtain the desired probability.

Exponential Approximation and Rare Event Estimates

When dealing with very small probabilities over a large number of independent trials (such as 80,000 couples), the computation can be simplified using the approximation 1 - (1 - p)^n ? 1 - e^(-np), where p is the probability of the event for one trial. This approach is particularly useful in estimating probabilities of rare events occurring at least once in a large sample.

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