Probability answers

1. What computation would you do to figure out for a group of 6 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 - P(all different) =

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

2. If I roll a 12-sided die 4 times, what is the probability that I get a different number every time?

P(all different) = (12/12)*(11/12)*(10/12)*(9/12)=57%

3. If I roll a 6-sided die 5 times, what is the probability that I get a 3 at least once

P(at least once) = 1-P(never) =

1-(5/6)(5/6)(5/6)(5/6)(5/6) = 1 - (5/6)^5=.598=60%

--OR--

P(never) =(5/6)(5/6)(5/6)(5/6)(5/6)=(5/6)^5=40% so

P(at least once) = 100%-40%=60%

4. If I roll a 10- sided die 3 times, what is the probability that I don't get a 2 on any of the rolls?

P(never 2) = (9/10)(9/10)(9/10)=(9/10)^3=93%

5. If I roll an 8 sided die 5 times, what is the probability that I get the same number (at least) twice?

P(same at least twice) = 1 - P(all different) = 1 - (8/8)(7/8)(6/8)(5/8)(4/8) = 79%

For the second quiz practice:

6. I have two 10-sided dice, numbered 1-10. If I toss both of them, the sums are shown on this chart:

One place to start is to make a grid: (grey is numbers on the dice)

(yes, I know this is a pretty big grid. I put it on the practice problems so that I could save the smaller grid problems for the quiz)

And mark evens, numbers over 10, equal to 10 and doubles (see my colored grids). Then count to find the probabilities

• a sum that is an even number? 50/100=1/2
• a sum that is over 10? 55/100=11/20
• a sum that is equal to 10? 9/100
• doubles? 10/100=1/10

7. I have a 10-sided die and a 6-sided die. If I toss them both, what is the probability that I will get:

• a sum that is at least 12?
• a sum that is over 10?
• a sum that is equal to 10?
• double

One place to start is to make a grid: (grey is numbers on the dice)

Then I can color the grid to show each situation and count.

• a sum that is at least 12? P=15/60=1/4
• a sum that is over 10? P=21/60=7/20
• a sum that is equal to 10? P=6/60=1/10
• double? P=6/60=1/10

8. I have two funny spotted 6-sided dice. Die A has the numbers: 1, 1, 3, 3, 5, 6 and die B has the numbers 2, 2, 2, 4, 4, 4

We can make a grid here too; in this grid I am marking which die would be larger:

• which die (if any) will be larger most often? Neither, they are each largest half of the time.
• What is the probability that the first die will be bigger than the second? 1/2

9. A.

Josh: 2, 2, 2, 2, 6, 6
Sarah:3, 3, 3, 3, 3, 3

Sarah has a higher number more often when playing against Josh

What is the probability Josh will have the higher number? 12/36=33%

B.

Sarah:3, 3, 3, 3, 3, 3
Kelly: 4, 4, 4, 4, 0, 0

Kelly wins more often when playing against Sarah

What is the probability that Sarah will have the higher number? 12/36=33%

C.

Kelly: 4, 4, 4, 4, 0, 0
Brian: 1, 1, 1, 5, 5, 5

Brian wins more often when playing against Kelly

What is the probability that Brian will have the higher number? 24/36=67%

D.

Brian: 1, 1, 1, 5, 5, 5
Josh: 2, 2, 2, 2, 6, 6

Josh wins most often against Brian

What is the probability that Brian will have the higher number? 12/36=33%

Put the dice in order so that person 1 beats person2, person 2 beats person 3, person3 beats person 4 but person 4 beats person 1 (see page 539#6)

Josh wins against Brian, who wins against Kelly, who wins against Sarah, (so you'd thing Josh's die was the best, and Sarah's was the worst except...) and Sarah wins against Josh. It kind of reminds me of an MC Escher drawing or maybe this one.

Josh: 2, 2, 2, 2, 6, 6
Brian: 1, 1, 1, 5, 5, 5
Kelly: 4, 4, 4, 4, 0, 0
Sarah:3, 3, 3, 3, 3, 3