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)

1 - (365/366)*(364/366)*(363/366)*(362/366)*(361/366)*(360/366)*(359/366)

(for 8 people, I have 7 fractions I am multiplying--for the 2nd, 3rd, 4th, 5th, 6th, 7th, and 8th person)

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

P(all different) = (11/12)(10/12)(9/12)(8/12)(7/12)(6/12)(5/12) = .05

P(all different) = 5%

1 - (11/12)(10/12)(9/12)(8/12)(7/12)(6/12)(5/12) = .95

P(2 in the same month) = 95%

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?

P(all different) = (5/6)*(4/6)*(3/6) = .28

P(all different) = 28%

1 - (5/6)*(4/6)*(3/6)= .72

P(2 the same number)= 72%

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?

P(all different) = (19/20)*(18/20)*(17/20)*(16/20) = .58

1 - (19/20)*(18/20)*(17/20)*(16/20) = .42

P(2 the same) = 42%

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

P(all different) = (7/8)(6/8) = .66

1 - (7/8)(6/8)

P(2 the same) = 34%

Notice that for all of these, there are one fewer fraction than there are people/dice/etc. This is because we want to know how many are different, and until we have at least two people, there is nothing to be different from. If you want to give the first person/die/etc. a fraction, you can give them the fraction 1=366/366=12/12=6/6=20/20=8/8. That's because they are guaranteed to not overlap with any previous person (because there isn't a previous person).