Expert Answer to "On each of three consecutive days the National..."

mattgondek4

Answered question

2021-09-26

On each of three consecutive days the National Weather Service announces that there is a 50-50 chance of rain. Assuming that the National Weather Service is correct, what is the probability that it rains on at most one of the three days? Justify your answer. (Hint: Represent the outcome that it rains on day 1 and doesn’t rain on days 2 and 3 as RNN.)

Answer & Explanation

mhalmantus

Skilled2021-09-27Added 105 answers

Step 1
As indicated by the query
There are only two possible outcomes for each day. Either it pours down rain or it doesn't.
So, there are use the binomial probability.
From the binomial probability,
$P (X = x) = n c_{x} \cdot P^{x} {(1 - P)}^{n - x}$
Now,
$n = 3, p = 0.5$
Here, need to find,
$N N N + R N N + N R N + N N R$
Step 2
Then, $P (X \leq 1) = P (X = 0) + P (X = 1)$
So,
$P (X = x) = n c_{x} \cdot P^{x} {(1 - P)}^{n - x}$
$P (X = 0) = 3 c_{0} \cdot {(0.5)}^{0} {(1 - 0.5)}^{3}$
$= 0.125$
Now,
$P (X = x) = n c_{x} \cdot P^{x} {(1 - P)}^{n - x}$
$P (X = 1) = 3 c_{1} \cdot {(0.5)}^{1} {(1 - 0.5)}^{2}$
$= 0.375$
Step 3
Now,
$P (X \leq 1) = P (X = 0) + P (X = 1)$
$= 0.125 + 0.375$
$= 0.5$
Thus, the probability that it rains on at most one of the three days is 0.5.

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