Multiplication Rule of Independent Events, Statistical Note 38

If an unbiased coin with two sides (Head and Tail) is tossed twice, what is the joint probability that head appears in both tosses?

The multiplication rule of probability of two independent events is the product of the probability of first independent marginal event and the probability of second independent marginal event. Same applies to the joint probability of multiple independent events.

Symbolically, if two events A and B are independent, the probability that both events and A and B occur is stated as:

P(A and B) = P(A)*P (B)

The events are said to be independent events if the occurrence of the first event does not affect the probability of the second and following events. In this example, occurrence of head or tail in the first toss does not affect the occurrence of head or tail in the second toss. Any of head or tail could appear in any number of toss of a coin. This applies to multiple tosses of a coin.

Let H be an independent event of turning head in a toss of a coin. Likewise, let T be an independent event of turning tail in a toss of a coin. The marginal probability of H, P(H) is one by two, 0.5. Similarly, the marginal probability of T, P(T) is one by two, 0.5.

The total number of outcomes in two tosses of a coin can be shown a tree diagram like the one shown below:

Figure 1: Marginal probabilities of independent events in first and second tosses of a coin and number of outcomes

Now, applying the multiplication rule of independent events, the joint probability that head appears in both tosses is the product of P(H) in the first toss and the P(H) in the second toss.

Symbolically, P(H and H) = ? * ? = ? = 0.25

Alternatively, the occurrence of head in both tosses is Outcome 1 of four outcomes in the tree diagram, and thus the probability is one out of four, equal to 0.25.