Consider a variation on the “random ice cream price” (pay what you roll) process: Two fair, six sided dice are rolled, but now the price is the smaller number followed by the larger number.

 

a. Determine the probability that you can afford the ice cream cone if you only have 25 cents.

Explain/justify your answer (here and throughout).

b. Determine the probability that the price is an odd number.

8. Consider another variation on the “random ice cream price” process: Two dice are rolled, but now these are 10-sided dice with the digits 0 – 9 on the sides. The price is back to being the larger number followed by the smaller number.

a. Determine the probability that the price is no more than 10 cents.

b. Determine the probability that the price is more than 50 cents.

 

 

9. Consider yet another variation on the “random ice cream price” process: Three fair, six sided dice are rolled, and the price is the largest number followed by the smallest number. If you are the customer and therefore prefer a smaller price, would you prefer to use two dice or three dice, or would you not have a preference? Explain your answer, but do not bother to perform any calculations.

10. Suppose that you encounter two traffic lights on your commute to school. Based on past experience, you judge that the probability is .45 that the first light will be red when you get to it, .35 that the second light will be red, and .25 that both lights will be red.

a. Produce a probability table to organize the given probabilities.

b. Determine the probability that at least one light will be red. Also name the relevant probability rule that you could use.

c. Determine the long-run percentage of days for which neither light will be red.

11. Consider that the 2012 U.S. Pet Ownership and Demographics Sourcebook reports that 36.5% of American households have a pet dog and 30.4% have a pet cat.

a. Does it follow from this information that 66.9% (the sum of 36.5% and 30.4%) of American households have a pet dog or a pet cat? Explain/justify your answer.

b. Based on the information given, what is the smallest possible value for the percentage of American households that have a pet dog or a pet cat?
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7

11 12 13 14 15 16
12 22 23 24 25 26
13 23 33 34 35 36
14 24 34 44 45 46
15 25 35 45 55 56
16 26 36 46 56 66

The table shows the prices resulting from the throws of the dice.

a. There are 36 cells in the table and 18 of them are 25 or less. So the probability is 18/36=1/2.

b. 15 prices are odd: probability is 15/36=5/12.

8

0 10 20 30 40 50 60 70 80 90
10 11 21 31 41 51 61 71 81 91
20 21 22 32 42 52 62 72 82 92
30 31 32 33 43 53 63 73 83 93
40 41 42 43 44 54 64 74 84 94
50 51 52 53 54 55 65 75 85 95
60 61 62 63 64 65 66 76 86 96
70 71 72 73 74 75 76 77 87 97
80 81 82 83 84 85 86 87 88 98
90 91 92 93 94 95 96 97 98 99

a. There are only three cells ≤10, so probability=3%, 0.03 or 3/100.

b. We can see from the table that there are 27 cells containg 50 or less. That means there are 73 cells containing more than 50. Therefore the probability=73%, 0.73 or 73/100.

9. The lowest price is 11 cents. With 2 dice this result has a probability of 1/36. With 3 dice for this result you would need to throw three ones. But the probability is much lower, 1/216. Therefore, the customer would prefer to use two dice.

10. The possible outcomes for two signals are: (row is first light, column is second light):

  2=Red (0.35) 2=Green (0.65)
1=Red (0.45) 0.25 0.45*0.65=0.2925
1=Green (0.55) 0.55*0.35=0.1925 0.55*0.65=0.3575

11a) There are 3 possibilities (assume that owning more than one dog or more than one cat counts as one dog or one cat):

i) dog(s) only

ii) dog(s) and cat(s)

iii) cat(s) only

36.5% covers i) and ii); 30.4% covers ii) and iii).

So there is an overlap of the sets of cat and dog owners.

We don't know how many households own both animals, so we cannot simply add the percentages.

11b) dog+both=36.5%; cat+both=30.4%, so dog-cat=6.1%; total=dog+cat+both=66.9%-both.

6.1+2cat=66.9-2both; 2cat=60.8-2both; cat=30.4-both. So if 30.4% owned a cat and a dog, there would be no one who owned only a cat. 6.1% would own only a dog, and the total would be 36.5% owning a dog or cat or both. All those owning a cat would also own a dog. But 6.1% would own a dog but not a cat.

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