What's the probability that in a randomly selected your of...

What's the probability that in a randomly selected your 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 initiate 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?

What's the probability that in a randomly selected your 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 initiate 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?

Key Concepts

Birthday Problem

The birthday problem is a famous probability puzzle that examines the likelihood of two or more people sharing the same birthday in a group. Despite there being 365 days in a year, the probability of a shared birthday reaches surprisingly high values even with a relatively small number of people.

Uniform Distribution Assumption

This concept involves assuming that each day of the year (excluding February 29) is equally likely for a birthday. This simplification transforms the real-world scenario into a uniform probability model, making the computations more straightforward and ensuring that each outcome (each day) has an equal chance.

Monte Carlo Simulation

Monte Carlo simulation is a technique that uses repeated random sampling to approximate a probability. In the context of the birthday problem, this involves generating random birthdays for each individual in the group using random digits, and then repeating the experiment multiple times to analyze the frequency of at least one shared birthday, thereby estimating the probability.

Empirical Versus Theoretical Probability

This concept distinguishes between theoretical probability, which is derived through analytical methods and assumptions (such as the 0.71 probability for shared birthdays in a group of 30), and empirical probability, which is obtained by actually simulating the process repeatedly and observing the outcomes. Comparing these two helps verify the accuracy of the theoretical predictions through practical experimentation.

Transcript

Please give Ace some feedback