Answers about complements:

1. a. If I spin 25 pennies, and 14 land heads up, what percentage of heads did I get? 56% (14 divided by 25 is .56)

b. What percentage tails did I get? (1-.56=.44) 44%

2. a. If I roll a normal die 20 times, and I get "6" 5 times, what percentage of the rolls were 6? (5/20=.25) 25%

b. What percentage of the rolls were less than 6? (100%-25%=75%) 75%

3. I have some red, blue and green beads in a bag. I did the following experiment 20 times: I took out three beads, and recorded if at least two were the same color. 70% of the time at least two were the same color. What happened the other 30% of the time? The other 30% of the time, they were all 3 different colors

4. I rolled two dice, and recorded whether I got the same number on both dice or not. 15% of the time, I got the same number on both dice. What happened the other 85% of the time? The other 85% of the time they were different numbers.

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:

5. 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)

6. In a group of 30, is it the probability more or less than 50% that (at least) two people will have the same birthday?

It is more than 50%. The probabilities exceed 50% when there are 23 people or more.

Variations:

7. 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%

8. In a group of 6 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) = .58

P(all different) = 58%

1 - (11/12)(10/12)(9/12)(8/12)(7/12)(6/12)(5/12) = .42

P(2 in the same month) = 42%