Question

Shameel has a flight to catch on Monday morning. His father will give him a ride to the airport. If it rains, the traffic will be bad and the probability that he will miss his flight is 0.06. If it doesn't rain, the probability that he will miss his flight is 0.01. The probability that it will rain on Monday is 0.19. Suppose that Shameel misses his flight. What is the probability that it was raining

Answer 1

The probability that it was raining on Monday given that Shameel misses his flight is 0.5846.

Explanation:

The Bayes' theorem states that the conditional probability of an event E
`_(i)`, of the sample space S = {E₁, E₂, E₃,...Eₙ}, given that another event A has already occurred is given by the formula:

`P(E_(i)|A)=\fracP(A{\sum\limits^(n)_(i=1) E_(i))P(E_(i))}`

Denote the events as follows:

X = it will rain on Monday

Y = Shameel misses his flight.

The information provided is:

`P(X) = 0.19\\P(Y|X)=0.06\\P(Y|X^(c))=0.01`

Compute the probability that it will not rain on Monday as follows:

`P(X^(c))=1-P(X)\\\\=1-0.19\\\\=0.81`

Compute the probability that it was raining on Monday given that Shameel misses his flight as follows:

Use the Bayes' theorem:

`P(X|Y)=(P(Y|X)P(X))/(P(Y|X)P(X)+P(Y|X^(c))P(X^(c)))`

`=((0.06* 0.19))/((0.06* 0.19)+(0.01* 0.81))\\\\=(0.0114)/(0.0114+0.0081)\\\\=(0.0114)/(0.0195)\\\\=0.58462\\\\\approx 0.5846`

Thus, the probability that it was raining on Monday given that Shameel misses his flight is 0.5846.