3.2 Independent and mutually exclusive events (modified r. bloom)

Independent events

Two events are independent if the following are true:

• $\mathrm{P(A|B)\; =\; P(A)}$
• $\mathrm{P(B|A)\; =\; P(B)}$
• $\mathrm{P(A AND B)\; =\; P(A) \cdot P(B)}$

If events $A$ and $B$ are independent , then the chance of $A$ occurring does not affect the chance of $B$ occurring and vice versa.

Translating the symbols into words, the first two mathematical statements listed above say that the probability for the event with the condition is the same as the probability for the event without the condition. For independent events: the condition does not change the probability for the event.

For example, two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the secondroll.

If you select 2 cards consecutively from a complete deck of playing cards, the two selections are not independent ; the result of the first selection changes the remaining deck and affects the probabilities for the second selection. This is referred to as selecting "without replacement"; the first card has not been replaced into the deck before the second card is selected.

However, suppose you were to select 2 cards "with replacement", by returning your first card to the deck and shuffling the deck before selecting the second card. Because the deck of cards is complete for both selections, the first selection does not affect the probability of the second selection. When selecting cards with replacement, the selections are independent .

To show that two events are independent, you must show only one of the conditions listed above. If any one of these conditions is true, then all of them are true.

Mutually exclusive events

Events $A$ and $B$ are mutually exclusive events if they cannot occur at the same time. This means that $A$ and $B$ do not share any outcomes and $\text{P(A AND B)}=0$ .

• For example, suppose the sample space $\mathrm{S\; =\; \{1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9,\; 10\}}$ .
• Let $\mathrm{A\; =\; \{1,\; 2,\; 3,\; 4,\; 5\},\; B\; =\; \{4,\; 5,\; 6,\; 7,\; 8\}}$ , and $\mathrm{C\; =\; \{7,\; 9\}}$ .
• $\text{A AND B}=\{4,5\}$ . $\text{P(A AND B) =}$ $\frac{2}{10}$ and is not equal to zero. Therefore, $A$ and $B$ are not mutually exclusive.
• $A$ and $C$ do not have any numbers in common so $\text{P(A AND C) = 0}$ . Therefore, $A$ and $C$ are mutually exclusive.

You must show that any two events are independent or mutually exclusive. You cannot assume either of these conditions.

If it is not known whether $A$ and $B$ are independent or dependent, assume they are dependent until you can show otherwise .

The following examples illustrate these definitions and terms.