9. Mutually Exclusive Events

Two or more events are said to be mutually exclusive if the occurrence of any one of them excludes the occurrence of the others (no common outcomes).

Thus if E1 and E2 are mutually exclusive events, then

P(E1 and E2) = 0.

Suppose "E1 or E2" denotes the event that "either E1 or E2 both occur", then

(a) If E1 and E2 are not mutually exclusive events:

P(E1 or E2) = P(E1) + P(E2) − P(E1 and E2)

We can also write:

P(E1 ∪ E2) = P(E1) + P(E2) − P(E1 ∩ E2)

(b) If E1 and E2 are mutually exclusive events:

P(E1 or E2) = P(E1) + P(E2)


EXAMPLE 1

It is known that the probability of obtaining zero defectives in a sample of 40 items is 0.34 whilst the probability of obtaining 1 defective item in the sample is 0.46. What is the probability of

(a) obtaining not more than 1 defective item in a sample?

(b) obtaining more than 1 defective items in a sample?


Answer


EXAMPLE 2

The probability that a student passes Mathematics is $\dfrac{2}{3}$ and the probability that he passes English is $\dfrac{4}{9}$. If the probability that he will pass at least one subject is $\dfrac{4}{5}$, what is the probability that he will pass both subjects?

(We assume it is based on probability only.)


Answer


Assorted Problems


EXAMPLE 3

A box contains 100 items of which 4 are defective. Two items are chosen at random from the box. What is the probability of selecting

(a) 2 defectives if the first item is not replaced;

(b) 2 defectives if the first item is put back before choosing the second item;

(c) 1 defective and 1 non-defective if the first item is not replaced?


Answer


EXAMPLE 4

Five small radios are packed in identical, unmarked individual sealed boxes. Three boxes are on table X and contain 2 radios made by firm A and one by firm B. Two boxes are on table Y and contain one radio made by firm A and one by firm B. If someone moves a box from table X to table Y and you randomly select a box from table Y, what is the probability that you will select a radio made by firm B?


Answer


EXAMPLE 5

If the independent probabilities that three people A, B and C will be alive in 30 years time are 0.4, 0.3, 0.2 respectively, calculate the probability that in 30 years' time,

(a) all will be alive

(b) none will be alive

(c) only one will be alive

(d) at least one will be alive


Answer




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