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Finding Probability with Given Restrictions

12. Let E, F, and G be three events. Find expressions for the events so that of E, F, and G:

(a) only E occurs;
(b) both E and G but not F occur;
(c) at least one of the events occurs;
(d) at least two of the events occur;
(e) all three occur;
(f) none of the events occurs.

Solution This solution is FREE courtesy of BrainMass!

Let the events E, F and G be independent of each other

Let the probability of occurrence of E be denoted by P(E)
Then the probability of E not occurring = Q(E) = {1-P(E)}

Let the probability of occurrence of F be denoted by P(F)
Then the probability of F not occurring = Q(F) = {1-P(F)}

Let the probability of occurrence of G be denoted by P(G)
Then the probability of G not occurring = Q(G) = {1-P(G)}

a) Only E occurs;

This means that E occurs and F and G do not occur
Probability of E occurring and F and G not occurring =
P(E) x Q (F) x Q (G)
= P(E) x {1-P (F)} x {1-P (G) }
Answer: P(E) x {1-P (F)} x {1-P (G) }

b) Both E and G but not F occur;

This means that E and G occur and F does not occur
Probability of E and G occurring and F not occurring =
P(E) x P (G) x Q (F)
= P(E) x P(G) {1-P (F)}

Answer: P(E) x P(G) {1-P (F)}

c) At least one of the events occurs;

Probability of at least one of the events occurring = 1- Probability that none of the events occur
Probability that none of the events occur means that E does not occur, F does not occur and G does not occur.
Thus probability that E does not occur, F does not occur and G does not occur = Q(E) x Q (F) x Q (G) = {1-P(E)} x {1-P (F)} x {1-P (G) }

Therefore probability that at least one of the event occurs
= 1- Q(E) x Q (F) x Q (G)
= 1-[ {1-P(E)} x {1-P (F)} x {1-P (G) }]
Answer: 1-[ {1-P(E)} x {1-P (F)} x {1-P (G) }]

d) At least two of the events occur;

There are 3 cases
Case 1: E and F occur , G does not occur
Probability = P(E) x P (F ) x Q (G) = P(E) x P (F ) x {1-P (G)}

Case 2: E and G occur , F does not occur
Probability = P(E) x P (G ) x Q (F) = P(E) x P (G ) x {1-P (F)}

Case 3: F and G occur , E does not occur
Probability = Q(E) x P (F ) x P (G) = {1-P(E)} x P (F ) x P (G)

Any one of the three may happen
Therefore probability that at least two of the events occur is the sum of probabilities for the three above cases=

P(E) x P (F ) x {1-P (G)} + P(E) x P (G ) x {1-P (F)} + {1-P(E)} x P (F ) x P (G)

Answer:
P(E) x P (F ) x {1-P (G)} + P(E) x P (G ) x {1-P (F)} + {1-P(E)} x P (F ) x P (G)

e) All three occur
This means that E, F and G all occur
Probability of E , F and G occurring
= P(E) x P (F) x P (G)

Answer: = P(E) x P (F) x P (G)
f) none of the events occurs.

Probability that none of the events occur means that E does not occur, F does not occur and G does not occur.
Thus probability that E does not occur, F does not occur and G does not occur =
Q(E) x Q (F) x Q (G)
= {1-P(E)} x {1-P (F)} x {1-P (G) }

Answer: = {1-P(E)} x {1-P (F)} x {1-P (G) }