Let a, b, c be events such that p(a) = 0.2 , p(b) = 0.3, p(c) = 0.4

find the probability that at least one of the events a and b occurs if

(a) a and b are mutually exclusive;
(b) a and b are independent.
find the probability that all of the events a, b, c occur if
(a) a, b, c are independent;
(b) a, b, c are mutually exclusive.

At least  one of the events A and B occurs:

(a) 0.5

(b) 0.44

All of the events A, B, C occur:

(a) 0.024

(b) 0

Step-by-step explanation:

Given:

P(A) = 0.2, P(B) = 0.3, P(C) = 0.4

At least  one of the events A and B occurs:

(a) If A and B are mutually exclusive events, then their intersection is 0.

Probability of at least one of them occurring means either of the two occurs or the union of the two events. Therefore,

(b) If A and B are independent events, then their intersection is equal to the product of their individual probabilities. Therefore,

All of the events A, B, C occur together:

(a) All of the events occurring together means the intersection of all the events.

If A, B, and C are independent events, then their intersection is equal to the product of their individual probabilities. Therefore,

(b) If A, B, and C are mutually exclusive events means events A, B, and C can't happen at the same time. Therefore, their intersection is 0.

d. 24+-0.114

step-by-step explanation:

the question requires us to calculate the approximate 95% confidence interval for the mean number of ounces of ketchup per bottle in the sample given the following statistics;

sample mean = 24

standard deviation, σ = 0.4

sample size , n = 49

the confidence interval for a population mean is calculated using the formula;

sample mean ± z-score*(σ/sqrt(n))

the z-score associated with a 95% confidence interval, from the standard normal tables, is;

1.96

substituting these values into the formula;

24 ± 1.96*(0.4/7)

= 24 ± 0.112

from the alternatives given, choice d is closest to the above expression;

step-by-step explanation: