Suppose the probability of a very cold day (temperature less than 0°C) on any random day of winter in a city is 0.15. If it is a very cold day, the probability of a car accident in the same city is 0.10. If it is not a very cold day, the probability of a car accident in the same city is 0.08. a) What is the probability that a random day is not a very cold day? b) What is the probability of no accident if it is not a cold day? c) What is the probability of an accident on a random day? d) Given that there was an accident today, what is the probability that today is a very cold day? e) Given that there was no accident today, what is the probability that today is not a very cold day? Part (a) Formula: P(C') = 1 − P(C) Part (b) Formula: P(NA|C') = 1 − P(A|C') Part (c) Formula: P(A) = P(A|C)P(C) + P(A|C')P(C') Part (d) Formula: P(C|A) = P(A|C)P(C) P(A) Part (e) Formulas: 1. Calculate P(NA) using the law of total probability: P(NA) = P(NA|C)P(C) + P(NA|C')P(C') 2. Then use Bayes' theorem: P(C'|NA) = P(NA|C')P(C') P(NA)
Suppose the probability of a very cold day (temperature less than 0°C) on any random day of winter in a city is 0.15. If it is a very cold day, the probability of a car accident in the same city is 0.10. If it is not a very cold day, the probability of a car accident in the same city is 0.08. a) What is the probability that a random day is not a very cold day? b) What is the probability of no accident if it is not a cold day? c) What is the probability of an accident on a random day? d) Given that there was an accident today, what is the probability that today is a very cold day? e) Given that there was no accident today, what is the probability that today is not a very cold day? Part (a) Formula: P(C') = 1 − P(C) Part (b) Formula: P(NA|C') = 1 − P(A|C') Part (c) Formula: P(A) = P(A|C)P(C) + P(A|C')P(C') Part (d) Formula: P(C|A) = P(A|C)P(C) P(A) Part (e) Formulas: 1. Calculate P(NA) using the law of total probability: P(NA) = P(NA|C)P(C) + P(NA|C')P(C') 2. Then use Bayes' theorem: P(C'|NA) = P(NA|C')P(C') P(NA)
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 4ECP: Show that the probability of drawing a club at random from a standard deck of 52 playing cards is...
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Probabilistic Methods
Transcribed Image Text:Suppose the probability of a very cold day (temperature less than 0°C) on any random day of winter in a city is 0.15. If it
is a very cold day, the probability of a car accident in the same city is 0.10. If it is not a very cold day, the probability of a
car accident in the same city is 0.08.
a) What is the probability that a random day is not a very cold day?
b) What is the probability of no accident if it is not a cold day?
c) What is the probability of an accident on a random day?
d) Given that there was an accident today, what is the probability that today is a very cold day?
e) Given that there was no accident today, what is the probability that today is not a very cold day?
Transcribed Image Text:Part (a)
Formula:
P(C') = 1 − P(C)
Part (b)
Formula:
P(NA|C') = 1 − P(A|C')
Part (c)
Formula:
P(A) = P(A|C)P(C) + P(A|C')P(C')
Part (d)
Formula:
P(C|A) =
P(A|C)P(C)
P(A)
Part (e)
Formulas:
1. Calculate P(NA) using the law of total probability:
P(NA) = P(NA|C)P(C) + P(NA|C')P(C')
2. Then use Bayes' theorem:
P(C'|NA) =
P(NA|C')P(C')
P(NA)
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