Multiplication Rule of Mutually Exclusive and Inclusive Events, Statistical Note 42

Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. In tossing of a coin, either of head or tail flips meaning they are mutually exclusive. An occurrence of head excludes an occurrence of tail in the same toss.

Let H be an event of turning head in a toss of a coin.  Likewise, let T be an 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 joint probability of head and tail in a single toss, denoted by P(H and T) is impossible, equal to zero.

Figure 1: Marginal probabilities in tossing of a coin

Likewise, in drawing a card from the deck of 52 cards, the occurrence of an ace card excludes cards of other ranks, two to ten, jack, queen and king. Let A be an event of drawing an ace card.  Likewise, let B be an event of drawing any non-ace card.  The marginal probability of A, P(A) is four by 52 or one by 13. Similarly, the marginal probability of any non-ace card, P(B) is 48 by 52 or 12 by 13. The joint probability of ace and any non-ace card in a single draw, denoted by P(A and B) is impossible, equal to zero.

Figure 2: Marginal probabilities in drawing a card from a deck

Mutually Inclusive Events: Two events are mutually inclusive or non-exclusive if the occur of an event does not exclude the occurrence of another event at the same time. In drawing a card from the deck of 52 cards, the occurrence of an ace card does not exclude the occurrence of a club card. Because, both identities - ace as a rank and club as a suit exist on the same card.

Let A be an event of drawing an ace card.  Likewise, let B be an event of drawing any non-ace card.  The marginal probability of A, P(A) is four by 52 or one by 13. Similarly, the marginal probability of any non-ace card, P(NA) is 48 by 52 or 12 by 13. Likewise, let C be an event of drawing a club card and NC be an event of drawing a non-club card. The joint probability of ace and club card in a single draw, denoted by P(A and C) is the product of P(A) and P(C), equal to 0.0192. Similarly, P(A and NC), P(NA and C) and P(NA and NC) are equal to 0.056, 0.2307 and 0.6923 respectively as shown in Figure 3.

Figure 3: Marginal probabilities of mutually exclusive events and joint probability of mutually inclusive events in a draw of a card from the deck