The Probability that a USA-based business
man goes to paris by car is 0.6 and by air is
0.4. If he goes by car, the probability that he
will be early for his appointment is 0.3 while if
he goes by air, the probability he'll be early is
0.65 (i) what is the probability that he goes early for
his appointment
(ii) One Monday morning, he arrived late for
his appointment. Find the probability that he
went by car
Where did the USA-based business man start his journey to Paris by car?
(i) the probability that he goes early for his appointment is 0.6x0.3 + 0.4x0.65
(ii) the probability that he went by car is (0.6x0.7)/(0.6x0.7+0.4x0.35)
To calculate the probabilities, we'll use the following formulas:
(i) P(Early) = P(Early|Car) * P(Car) + P(Early|Air) * P(Air)
(ii) P(Car|Late) = (P(Late|Car) * P(Car)) / P(Late)
Given:
P(Car) = 0.6 (probability of going to Paris by car)
P(Air) = 0.4 (probability of going to Paris by air)
P(Early|Car) = 0.3 (probability of being early if going by car)
P(Early|Air) = 0.65 (probability of being early if going by air)
P(Late) = ?
(i) To find the probability of going early, we substitute the given values into the formula:
P(Early) = P(Early|Car) * P(Car) + P(Early|Air) * P(Air)
= 0.3 * 0.6 + 0.65 * 0.4
= 0.18 + 0.26
= 0.44
Therefore, the probability that he goes early for his appointment is 0.44 or 44%.
(ii) To find the probability that he went by car given that he arrived late, we'll use Bayes' theorem:
P(Car|Late) = (P(Late|Car) * P(Car)) / P(Late)
To calculate P(Late), we can use the complement rule:
P(Late) = 1 - P(Early)
Substituting the given values and calculated probabilities, we have:
P(Car|Late) = (P(Late|Car) * P(Car)) / P(Late)
= (1 - P(Early|Car)) * P(Car) / (1 - P(Early))
Using the values we calculated earlier, we get:
P(Car|Late) = (1 - 0.3) * 0.6 / (1 - 0.44)
= 0.7 * 0.6 / 0.56
= 0.42 / 0.56
= 0.75
Therefore, the probability that he went by car given that he arrived late is 0.75 or 75%.