Supersonics Records plans to market a new record for a comeback artist nationwide. Management believes that there is a 70% chance that the new record will be a hit and, therefore, a 30% chance that the new record will be a miss. However, before resources are committed to market this new record nationwide, management will ask Mr. Deejay, a well-known authority in the music scene, to predict whether the new record will be a hit or a miss. Mr. Deejay's past predictions were quite impressive-che correctly predicted a hit record 80% of the time and incorrectly predicted a hit record only 10% of the time. Let H = The new record is actually a hit M = The new record is actually a miss DH = Mr. Deejay predicts the new record will be a hit Therefore, P(H) = 0.70 P(M) = 0.30 P(DH/H) = 0.80 P(DH/M) = 0.10 Based on these information, what is P(H and DH)? (That is, what is the probability that the new record is actually a hit (H) AND Mr. Deejay predicts the new record will be a hit (DH)?)
A. 0.56
B. 0.59
C. 0.95
D. Not enough information given to answer this question
E. None of the above

Answer:

A. 0.56

Step-by-step explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which

P(B|A) is the probability of event B happening, given that A happened.

[tex]P(A \cap B)[/tex] is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

We want to find [tex]P(H \cap DH)[/tex].

We can relate the numbers using [tex]A = H, B = DH[/tex]. So

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

[tex]P(DH|H) = \frac{P(DH \cap H)}{P(H)}[/tex]

P(H) = 0.70 , P(DH/H) = 0.80

So

[tex]P(DH \cap H) = P(DH|H)*P(H) = 0.8*0.7 = 0.56[/tex]

The correct answer is given by option A.

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