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1) Frank wants to earn as many points as possible in one turn in a game. Two number cubes whose sides are numbered 1 through 6 are rolled. He is given two options for the manner in which points are awarded in the turn.

OPTION A: If the sum of the rolls is a prime number, Frank receives 12 points.

OPTION B: If the sum of the rolls is a multiple of 4, Frank receives 24 points.

Which statement best explains the option he should choose?

 

A) Frank should choose Option A. The mathematical expectation of this option is 6 and is the greater mathematical expectation of the two options.

B) Frank should choose Option B. The mathematical expectation of this option is 5 and is the greater mathematical expectation of the two options.

C) Frank should choose Option B. The mathematical expectation of this option is 6 and is the greater mathematical expectation of the two options.

D) Frank should choose Option A. The mathematical expectation of this option is 5 and is the greater mathematical expectation of the two options.

 

2) Danielle may choose one of two options for the method in which she may be awarded a money prize.

OPTION A: Spin a spinner twice. The spinner is divided into four equally-sized sectors numbered 1, 4, 4, and 5. If the sum of the two spins is greater than 6, Danielle is awarded $8. Otherwise, she must pay $2.

OPTION B: Flip a coin three times. If heads appears once, Danielle is awarded $6. Otherwise, she must pay $1.

Danielle chooses the option with the greater mathematical expectation.

How much more money can Danielle expect to make by choosing this option over the other option?

 May 2, 2019

Best Answer 

 #1
avatar+110 
+2

1) Considering Option A: prime numbers up to 12 are 2, 3, 5, 7, and 11. The probability of obtaining these is \(\frac{15}{36}\) (sum probabilities: P(2) + P(3) + P(5) + P(7) + P(11) = \(\frac{1}{36} + \frac{2}{36} + \frac{4}{36} + \frac{6}{36} + \frac{2}{36}\)). Multiplying the probability of obtaining a prime number by the given point value of 12 gives us 5 as the expected value of Option A.

Considering Option B: multiples of 4 up to 12 are 4, 8, and 12. The probability of obtaining these is \(\frac{9}{36}\) (sum probabilities: P(4) + P(8) + P(12) = \(\frac{3}{36} + \frac{5}{36} +\frac{1}{36}\)). Multiplying the probability of obtaining a multiple of 4 by the given point value of 24 gives us 6 as the expected value of Option B.

Therefore, the answer is C with Option B and the expectation being 6.

Here's a good table for two-dice sum probabilities: http://lh4.ggpht.com/-RSwGhQhT9JI/UblGzz7HNvI/AAAAAAAABRA/8mTpypdmFZA/TwoDiceTable9.jpg?imgmax=800

 

2) Considering Option A: Sum chart: 

  1 4 4 5
1 2 5 5 6
4 5 8 8 9
4 5 8 8 9
5 6 9 9 10

So, \(P(2) = \frac{1}{16}, P(5) = \frac{4}{16}, P(6) = \frac{2}{16}, P(8) = \frac{4}{16}, P(9) = \frac{4}{16}, P(10) = \frac{1}{16}\). Then, \(P(sum > 6) = \frac{4}{16} + \frac{4}{16} + \frac{1}{16} = \frac{9}{16}\) and \(P(sum \leq 6) = 1 - \frac{9}{16} = \frac{7}{16}\). Multiplying the winning probability of \(\frac{9}{16}\) by $8, we get an expected value of $4.50. Multiplying the losing probability of \(\frac{7}{16}\) by $2, we get an expected value of $0.875. Subtracting these two expected values, our overall expectation is $3.625.

 

Considering Option B: the probability of flipping a coin three times and heads appearing exactly once is \(\frac{3}{8}\). The probability of losing is then \(1 - \frac{3}{8} = \frac{5}{8}\). Multiplying the probability of winning \(\frac{3}{8}\) by $6 gives us an expected value of $2.25. Multplying the probability of losing \(\frac{5}{8}\) by $1 gives us an expected value of $0.625. Subtracting these two expected values, our overall expectation is $1.625.

Good display of flipping a coin three times results: http://web.mnstate.edu/peil/MDEV102/U3/S25/Cartesian3.PNG

 

Therefore, Danielle would choose Option A. Danielle would expect to make $2 more choosing Option A over Option B.

 May 2, 2019
 #1
avatar+110 
+2
Best Answer

1) Considering Option A: prime numbers up to 12 are 2, 3, 5, 7, and 11. The probability of obtaining these is \(\frac{15}{36}\) (sum probabilities: P(2) + P(3) + P(5) + P(7) + P(11) = \(\frac{1}{36} + \frac{2}{36} + \frac{4}{36} + \frac{6}{36} + \frac{2}{36}\)). Multiplying the probability of obtaining a prime number by the given point value of 12 gives us 5 as the expected value of Option A.

Considering Option B: multiples of 4 up to 12 are 4, 8, and 12. The probability of obtaining these is \(\frac{9}{36}\) (sum probabilities: P(4) + P(8) + P(12) = \(\frac{3}{36} + \frac{5}{36} +\frac{1}{36}\)). Multiplying the probability of obtaining a multiple of 4 by the given point value of 24 gives us 6 as the expected value of Option B.

Therefore, the answer is C with Option B and the expectation being 6.

Here's a good table for two-dice sum probabilities: http://lh4.ggpht.com/-RSwGhQhT9JI/UblGzz7HNvI/AAAAAAAABRA/8mTpypdmFZA/TwoDiceTable9.jpg?imgmax=800

 

2) Considering Option A: Sum chart: 

  1 4 4 5
1 2 5 5 6
4 5 8 8 9
4 5 8 8 9
5 6 9 9 10

So, \(P(2) = \frac{1}{16}, P(5) = \frac{4}{16}, P(6) = \frac{2}{16}, P(8) = \frac{4}{16}, P(9) = \frac{4}{16}, P(10) = \frac{1}{16}\). Then, \(P(sum > 6) = \frac{4}{16} + \frac{4}{16} + \frac{1}{16} = \frac{9}{16}\) and \(P(sum \leq 6) = 1 - \frac{9}{16} = \frac{7}{16}\). Multiplying the winning probability of \(\frac{9}{16}\) by $8, we get an expected value of $4.50. Multiplying the losing probability of \(\frac{7}{16}\) by $2, we get an expected value of $0.875. Subtracting these two expected values, our overall expectation is $3.625.

 

Considering Option B: the probability of flipping a coin three times and heads appearing exactly once is \(\frac{3}{8}\). The probability of losing is then \(1 - \frac{3}{8} = \frac{5}{8}\). Multiplying the probability of winning \(\frac{3}{8}\) by $6 gives us an expected value of $2.25. Multplying the probability of losing \(\frac{5}{8}\) by $1 gives us an expected value of $0.625. Subtracting these two expected values, our overall expectation is $1.625.

Good display of flipping a coin three times results: http://web.mnstate.edu/peil/MDEV102/U3/S25/Cartesian3.PNG

 

Therefore, Danielle would choose Option A. Danielle would expect to make $2 more choosing Option A over Option B.

Anthrax May 2, 2019
 #2
avatar+100586 
+1

Nice, Anthrax.....!!!!!

 

 

cool cool cool

 May 3, 2019

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