contestada

Bryan records the number of hours he sleeps each night for several days and whether it is raining in the morning when he
wakes up. Bryan concludes that these two events are independent:
• Bryan sleeps 8 or more hours.
• It is raining in the morning.
Based on Bryan's conclusion, which statement must be true?
Bryan never sleeps 8 or more hours on days that it is not raining in the morning.
The probability that Bryan sleeps 8 or more hours is the same whether or not it is raining in the morning.
The probability that Bryan sleeps 8 or more hours is influenced by whether or not it is raining in the morning.
The probability that Bryan sleeps 8 or more hours is the same as the probability that it is raining in the morning.

Respuesta :

Using conditional probability, it is found that the correct statement is:

The probability that Bryan sleeps 8 or more hours is the same whether or not it is raining in the morning.

What is Conditional Probability?

Conditional probability is the probability of one event happening, considering a previous event. The formula 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 problem, the events are as follows:

  • Event A: Bryan sleeps 8 or more hours.
  • Event B: It is raining in the morning.

The events are independent, hence:

[tex]P(A \cap B) = P(A)P(B)[/tex]

Then:

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

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

Hence the correct option is:

The probability that Bryan sleeps 8 or more hours is the same whether or not it is raining in the morning.

More can be learned about conditional probability at https://brainly.com/question/14398287