The probability Barry is either on time for school or there is no sick
passenger on the train is 8/9. The probability Barry will be late for school if
there’s a sick passenger on the train is 2/3. Find the probability
(a)Barry is late for school and there is a sick passenger on the train.
(b) there is a sick passenger on the train.
is part a 1/9 * 2/3? and im not sure what to do for b
To find the probability for each scenario described, we can assign variables to the events and use the given information to form equations.
Let's define the events:
A: Barry is late for school
B: There is a sick passenger on the train
We are given the following probabilities:
P(A ∪ ¬B) = 8/9 (the probability Barry is either on time for school or there is no sick passenger on the train)
P(A | B) = 2/3 (the probability Barry will be late for school given that there's a sick passenger on the train)
(a) We want to find P(A and B) (the probability Barry is late for school and there is a sick passenger on the train).
Using the formula for conditional probability, we know:
P(A and B) = P(B) * P(A | B)
We are not given P(B) directly, but we can solve for it using the equation:
P(A ∪ ¬B) = P(B) + P(¬B) - P(B and ¬A)
P(A ∪ ¬B) = P(B) + (1 - P(B)) - P(B and ¬A)
8/9 = P(B) + (1 - P(B)) - P(B) * P(¬A)
Simplifying:
8/9 = 1 - P(B) * P(¬A)
Now let's use the given information:
P(A | B) = 2/3
From the conditional probability formula, we have:
P(A | B) = P(A and B) / P(B)
Substituting the known values:
2/3 = (P(A and B)) / P(B)
We can rewrite this as:
P(A and B) = (2/3) * P(B)
Now we have two equations:
8/9 = 1 - P(B) * P(¬A)
P(A and B) = (2/3) * P(B)
We can solve these equations together to find the values.
(b) To find the probability that there is a sick passenger on the train (P(B)), we can use the equation we derived earlier:
P(A ∪ ¬B) = P(B) + (1 - P(B)) - P(B) * P(¬A)
Substituting the known values:
8/9 = P(B) + (1 - P(B)) - P(B) * P(¬A)
We can solve this equation to find P(B).