Question

Two friends play a silly game where one person randomly strikes a key on a piano and the other person guesses the pitch of the note that was played. There are 12 different pitches represented on the piano, and since neither friend is that musical, their guesses of pitch are equally likely. Additionally, there are 5 black keys for every 7 white keys, and because the white keys are bigger they are more likely to be hit.

a.)(8 points)Assume that the probability of hitting a white key is P(W)=14/19 and the probability of hitting a black key is P(B)=5/19. Assuming successive strikes are independent, what is the probability that 5 white keys are struck in a row? The probability that 5 black keys are struck in a row?

b.)(5 points)Let GW be the event of guessing a pitch that belongs to a white key, and GB the event of guessing a pitch that belongs to a black key. What are the probabilities P(GW) and P(GB)?

c.)(12 points)It’s safe to assume that the guess and the key strike are independent of one another since neither person playing is good at recognizing pitch. What are the following probabilities:P(GW∩W)P(GB∩B)P(GW∪B)P(GB∪W)P(GW∪W)P(GB∪B).

Answer 1

Answer:-

Given That:-

Two friends play a silly game where one person randomly strikes a key on a piano and the other person guesses the pitch of the note that was played. There are 12 different pitches represented on the piano, and since neither friend is that musical, their guesses of pitch are equally likely. Additionally, there are 5 black keys for every 7 white keys, and because the white keys are bigger they are more likely to be hit.

a.)(8 points)Assume that the probability of hitting a white key is P(W)=14/19 and the probability of hitting a black key is P(B)=5/19. Assuming successive strikes are independent, what is the probability that 5 white keys are struck in a row? The probability that 5 black keys are struck in a row?

Given,

P(W) = 14/19

P(B) = 5/19

Both are assumed in part (a)

strikes are independent

P(5 white keys stuck in a row) = 14/19 * 14/19 * 14/19 * 14/19 * 14/19

= 537824/2476099

=0.2172

P(5 black keys stuck in a row) = 5/19 * 5/19 * 5/19 * 5/19 * 5/19 * 5/19

= 3125/2476099

= 0.00126

0.0013

b.)(5 points)Let GW be the event of guessing a pitch that belongs to a white key, and GB the event of guessing a pitch that belongs to a black key. What are the probabilities P(GW) and P(GB)?

P(GW) = 1/2

P(GB) = 1/2

The guesses of pitch are equally likely.

c.)(12 points)It’s safe to assume that the guess and the key strike are independent of one another since neither person playing is good at recognizing pitch. What are the following probabilities:P(GW∩W)P(GB∩B)P(GW∪B)P(GB∪W)P(GW∪W)P(GB∪B).

P(GW∩W)= 1/2 * 7/12

= 7/24

P(W) = 7/12

P(B) = 5/12

P(GB∩B) = 1/2 * 5/12

= 5/24

P(GW∪B) = P(GW) + P(B) - P(GW ∩ B)

= 1/2 + 5/12 - 5/24

= (12+10-5)/24

= 17/24

P(GB∪W) = P(GB) + P(W) - P(GB ∩ W)

= 1/2 + 7/12 - 7/24

= (12+14-7)/24

= 19/24

P(GW∪W) = P(GW) + P(W) - P(GW ∩ W)

= 7/12 + 1/2 - 7/24

= 19/24

P(GB∪B).= P(GB) + P(B) - P(GW ∩ B)

= 1/2 + 5/12 - 5/24

= 17/24

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