Answers:
- 9. independent; 8.3%
- 10. dependent; 6.4%
- 11a. independent; 12.5%
- 11b. dependent; 10.3%
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Explanation for Problem 9
The events are independent because any given coin flip doesn't affect the other one. Also the coin flips do not affect the dice roll, or vice versa.
1/2 = probability of tails
(1/2)^2 = 1/4 = probability of two tails in a row
2/6 = 1/3 = probability of rolling a '5' or '6'
(1/4)*(1/3) = 1/12 = 0.083 = 8.3% is the approximate probability of two tails and getting a dice roll larger than 4.
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Explanation for Problem 10
These events are dependent. Why? It's solely because we do not put the coin back when selecting it. This is a case of "no replacement".
There are 7+2+5 = 14 coins total. And 7/14 = 1/2 is the probability of getting a quarter on the 1st selection.
After that coin is not put back, there are 14-1 = 13 coins left. Of which 5 are nickels. Therefore 5/13 is the probability of getting a nickel on the 2nd selection. Then 4/12 = 1/3 is the probability of getting another nickel on the 3rd selection.
Multiply those fractions
(1/2)*(5/13)*(1/3) = 5/78 = 0.064 = 6.4% is the final answer. It is approximate.
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Explanation for Problem 11, part a)
Each selection is replaced, so the probability of the next selection is not altered. Each event is independent.
9 yellow out of 4+9+5 = 18 total
9/18 = 1/2 = probability of getting yellow
(1/2)^3 = 1/8 = 0.125 = 12.5% = probability of 3 yellow in a row when replacement is done.
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Explanation for Problem 11, part b)
We aren't doing replacement. The probability will change as we select more marbles. The events are dependent.
9 yellow and 4+9+5 = 18 total
- 9/18 = probability of getting yellow on 1st selection
- 8/17 = probability of getting yellow on 2nd selection
- 7/16 = probability of getting yellow on 3rd selection
Each numerator counts down by one (9,8,7) and so do the denominators (18,17,16).
Those multiply to (9/18)*(8/17)*(7/16) = 7/68 = 0.103 = 10.3% is the approximate final answer.