Multiplication rule for independent events

Two events A and B are independent if

Equivalent conditions are or .

Example:

An urn contains three red balls, two blue balls, and five white balls. A ball is selected and its colour is noted. Then it is replaced. A second ball is selected and its colour is noted. Find the probability of

a) Selecting two blue balls

b) Selecting a blue and then white ball

c) Selecting a red ball and then a blue ball

Solution:

a) P (blue and blue)= P (blue)· P (blue)=

b) P (blue and white)= P (blue)· P (white)=

c) P (red and blue)= P (red)· P (blue)=

Example:

An urn contains five red and seven blue balls. Suppose that two balls are selected at random with replacement. Let A and B be the events that the first and the second balls are red, respectively. Then we get . Now since and .

Thus A and B are independent.

If we do the same experiment without replacement, then

while

as expected. Thus implying that A and B are dependent.

Remark:

Multiplication rule for independent events can also be extended to three or more independent events by using the formula

.

Example:

The probability that a specific medical test will show positive is 0.32. If four people are tested, find the probability that all four will show positive.

Solution:

Let (i =1, 2, 3, 4) be the symbol for a positive test result.

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