Probability questions # 2

The Birthday problem and similar stuff

Important ideas in the birthday problem:

• You should start with finding the probability that everyone has a different birthday.
• You find the probability that the birthdays are different, by thinking about people 1 by 1:
  • What is the probability that the second person's birthday is different than the first person? (notice that I start with the second person because I know that you can't have two birthdays the same if there is only one person)
  • If the first two are different, what is the probability that the third person's birthday is different from the first two?
  • If the first 3 are different, wha is the probability that the fourth person's birthday is different from those 3 days?
  • etc.
• The complement of everyone having a different birthday is (at least) two people having the same birthday.
• So, the P(two people have the same birthday) = 1-P(everyone's birthdays are different)

Some problems:

The birthday problem itself:

1. What computation would you do to figure out for a group of 8 people what the probability is that two people have the same birthday? (just write out the mulitplication, don't crunch the numbers) (assume 366 equally likely days per year)

Variations:

2. In a group of 8 people, what is the probability that two people's birthdays are in the same month?

3. If everyone in a group of 4 people rolls a 6 sided die, what is the probability that (at least) two people roll the same number?

4. If everyone in a group of 5 people rolls a 20 sided die, what is the probability that (at least) two people roll the same number?

5. If I roll 3 8-sided dice, what is the probability that (at least) two of the dice will have roll same number?

Answers