Independent and Mutually Exclusive Events

Consider the task of tossing a die and flipping a coin at the same time. Let Hn denote that the coin landed on heads and n was tossed on the die and let Tn denote that the coin landed on tails and n was tossed on the die.

The sample space for this experiment is

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Let A be the event that an even number was tossed on the die.
Let B be the event that a number divisible by 5 was tossed on the die.
Let C be the event that the coin landed on heads.

Let's represent these events as sets.

A = {H2, H4, H6, T2, T4, T6}
B = {H5, T5}
C = {H1, H2, H3, H4, H5, H6}

P(A) = 6/12 = 1/2
P(B) = 2/12 = 1/6
P(C) = 6/12 = 1/2

Since there is no number between 1 and 6 that is both even and divisible by 5 the events, A and B, cannot take place simultaneously. A and B are said to be mutually exclusive events or in other words their sets do not intersect.

The probability that A or B takes place may be calculated as follows:
P(A or B) = P(A) + P(B) = 1/2 + 1/6 = 4/6

Since it is possible for an even number to be tossed on the die and for the coin to land on heads simultaneously then A and C are not mutually exclusive. In fact A ∩ C is a non-empty set,  A ∩ C = {H2, H4, H6}.

Two events are considered independent when the outcome of one does not influence the outcome of the other. In other words, the success of one event does not affect the probability of the other event. A and C are independent events and B and C are independent events since the side the coin lands on is in no way affected by the number tossed on the die.

Since A and C are independent, the probability that A and C take place simultaneously may be calculated as follows:
P(A and C) = P(A) × P(C) = 1/2 + 1/2 = 1/4

How do we calculate the probability of A or C? Since A and C are independent we use the following formula:
P(A or C) = P(A) + P(C) - P(A and C) = 1/2 + 1/2 - 1/4 = 3/4


Test yourself, are events B and C mutually exclusive? How would you calculate P(B or C)?