Question

Q1. Suppose that four regular 6-sided dice are flipped.

1.What is the probability of a "three-of-a-kind" that is, three of the dice show the same number and the other die shows a different number?

2. What is the probability that at least two of the dice show the same number?

Answer 1

Hi there! I'd like to help you understand how to do this.

Answer 2

yes please

Answer 3

you there

Answer 4

OK, let's start a little simpler, what if there are only two dice?

Answer 5

Can you compute the probability they show the same answer?

Answer 6

less probability

Answer 7

well how can we quantify it? Hint: the first die will be some number, it doesn't matter what; what's the probability the second die comes up the same?

Answer 8

(1,1) (2.2) and so on
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Answer 9

well those are possible outcomes, but we're trying to compute probability. Let me try another way, what's the probability, if you roll a single die, that it comes up with a 1?

Answer 10

1/6

Answer 11

great. so if you have two dies, it doesn't matter what the first one is -- what's the probability that the second will be the same?

Answer 12

two die 2/12

Answer 13

well, yes, although i'm not sure you got there quite right - basically the first die will have some value. no matter what that value is, the second die will have a 1/6 chance of coming up the same. does that make sense?

Answer 14

pardon

Answer 15

yes?

Answer 16

what do you mean

Answer 17

okay, you roll the first die, it comes up with some number -- 4, 2, 1, it doesnt matter what. then you roll the second die. what's the probability that this second die will come up the same as the first?

Answer 18

1/6

Answer 19

i guess

Answer 20

two separate occassion

Answer 21

nice! Okay so let's do three now. You roll the first die. It comes up, say, a 3. What's the probability the second AND third dice both come up a 3?

Answer 22

did i say correct bfr

Answer 23

3/18

Answer 24

nope. two events that are independent, you multiply the probabilities.

Answer 25

2/12

Answer 26

let me try this a different way. Imagine you roll two dice, what's the probability that the first one comes up 5, AND, the second one comes up 2?

Answer 27

5/6 , 2/6

Answer 28

are you saying the probability of a die coming up with 5 is different than the probability it will come up as, say, a 4?

Answer 29

can u cvlear the concept you r mkin more complicated for me i guess

Answer 30

well everything we're talking about so far is much simpler than the actual question you asked

Answer 31

ok, let me try this.. if you roll a single die, what's the probability that what will come up is a 5?

Answer 32

1/6

Answer 33

OK, and what's the probability that what will come up is a 2?

Answer 34

2/6

Answer 35

why 2/6?

Answer 36

the probability of any number coming up is the same, right?

Answer 37

ya ohh yes so then

Answer 38

okay so what's the probability of a die coming up 2?

Answer 39

1/6

Answer 40

would be same

Answer 41

ok, and a probability of it coming up a 5?

Answer 42

1/6

Answer 43

great. so what's the probability that we roll two at the same time, and the first one comes up 2, and the second comes up 5?

Answer 44

2/12

Answer 45

why?

Answer 46

outcome will double

Answer 47

2/12 is the same as 1/6 isn't it?

Answer 48

and 6+6

Answer 49

what do you mean by "outcome wil double"?

Answer 50

6+6

Answer 51

no. If two things happen independently, you *multiply* the probability of the individual events to find the probability that they both happen.

Answer 52

2/36

Answer 53

closer. you're multiplying 1/6 times 1/6

Answer 54

what's 1/6 times 1/6?

Answer 55

why

Answer 56

you said 2/36, that's not correct

Answer 57

if two events happen independently, then the probability that they *both* happen is the individual probabilities, multiplied.

Answer 58

then

Answer 59

2/12

Answer 60

so if one die has a 1/6 chance of coming up 2, and a second die separately has a 1/6 chance of coming up 5, then the probability that this both happens -- i.e. that you roll two dice, and they come up 2 and 5, is....?

Answer 61

OK, can you do the following math problem:
\[{ 1 \over 6} x {1 \over 6 }\]

Answer 62

it is not 2/12 and not 2/36

Answer 63

1/36

Answer 64

is it ok now

Answer 65

OK, yes. Nice. Now, what's the probability if you roll *three* dice, that the first comes up 5, and second comes up 2, and the third comes up 4?

Answer 66

1/216 i guess if iam not wrong

Answer 67

yes, you got it!

Answer 68

ok now we're going to do something a little different. What's the probability you roll a single die, and it comes up with any number *except* for 4?

Answer 69

we're getting closer to your final answer

Answer 70

1/5

Answer 71

nope

Answer 72

confusing

Answer 73

1/6-1/4

Answer 74

is it ok now

Answer 75

r u there

Answer 76

helllo

Answer 77

hi sorry

Answer 78

what's the probability that 1, 2, 3, 4, or 6 comes up? i.e., any number except 5? there are 5 ways this can happen out of the 6 ways a die can come up.

Answer 79

hello?

Answer 80

hello

Answer 81

r u there now