The birthday problem What’s the probability that in a randomly selected group of$30$ unrelated people, at least two have the same birthday? Let’s make two

assumptions to simplify the problem. First, we’ll ignore the possibility of a February $29$ birthday. Second, we assume that a randomly chosen person is equally likely to be born on each of the remaining $365$ days of the year.

(a) How would you use random digits to imitate one repetition of the process? What variable would you measure?

(b) Use technology to perform $5$ repetitions. Record the outcome of each repetition.

(c) Would you be surprised to learn that the theoretical probability is $0.71$? Why or why not?

$n=30$ persons were chosen at random. Assume that a randomly picked person is equally likely to be born on each of the other $365$ days of the year, ignoring the probability of a February $29$ birthday.

We can't foresee the outcomes of a chance process, yet they have a regular distribution over a large number of repetitions. According to the law of large numbers, the fraction of times a specific event occurs in numerous repetitions approaches a single number. The likelihood of a chance outcome is its long-run relative frequency. A probability is a number between $0$ (never happens) and $1$ (happens frequently) (always occurs).

To simulate one repetition of the procedure, use random digits as follows: Use numbers with three digits. It corresponds to a birthday if the numbers are between $000$ and $364$ If the number is in the range of $365$ to $999$ it should be ignored. Randomly select numbers between $000$ and $364$ Then count how many times each number appears in the $30$-number list. The number of times a number appears is counted.

Use numbers with three digits. It corresponds to a birthday if the numbers are between $000$and $364$If the number is in the range of $364$ to $999,$ it should be ignored. Randomly select numbers between $000$ and $999$ For one simulation, execute the following command on your $Ti83/Ti84$ calculator: $rand\ln t(0,364,30)$, where the command $rand\ln t$ is found under $PRB$ in the MATH-menu. Five trips through the simulation. Determine the number of times each simulation is repeated.

The theoretical probability of $0.71$ is not surprising, given that (almost) all simulations had repeated birthdays in exercise (b). As a result, it's not surprising.