What Is Probability?

What Is Probability? To discuss probability, let’s begin by defining some terms. An experiment is a process, such as tossing a coin, that gives definite results, called the outcomes of the experiment. The sample space of an experiment is the set of all possible outcomes. If we let H stand for heads and T for tails, then the sample space of the coin-tossing experiment is S = {H, T }.

What Is Probability? We will be concerned only with experiments for which all the outcomes are equally likely. For example, when we toss a perfectly balanced coin, heads and tails are equally likely outcomes in the sense that if this experiment is repeated many times, we expect that about as many heads as tails will show up.

What Is Probability? Another way of saying this is that probability is: Number of ways to get success / Total possible outcomes

What Is Probability? The closer the probability of an event is to 1, the more likely the event is to happen; the closer to 0, the less likely. If P (E) = 1, then E is called a certain event; if P (E) = 0, then E is called an impossible event. Notice that the probability P (E) of an event is a number between 0 and 1, that is, 0 £ P (E) £ 1

Calculating Probability by Counting

Calculating Probability by Counting To find the probability of an event, we do not need to list all the elements in the sample space and the event. We need only the number of elements in these sets. EX. A five-card poker hand is drawn from a standard deck of 52 cards. What is the probability that all five cards are spades?

Example 3 – Finding the Probability of an Event Solution: The experiment here consists of choosing five cards from the deck, and the sample space S consists of all possible five-card hands. We need to first calculate the number of ways to pull 5 spades from a deck and divide by the total number of ways to choose any 5 cards from a deck.

Example 3 – Solution cont’d Thus the number of elements in the sample space is The event E that we are interested in consists of choosing five spades. Since the deck contains only 13 spades, the number of ways of choosing five spades is

Example 3 – Solution Thus the probability of drawing five spades is cont’d Thus the probability of drawing five spades is

EX. 2 An experiment consists of tossing a coin and rolling a die. Find the sample space. Find the probability of getting heads and an even number Find the probability of getting heads and a number greater than 4 Find the probability of getting tails and an odd number.

EX. 3 A committee of 5 people is to be formed from a group of 10 women and 6 men. Find the probability that: 3 women and 2 men are chosen 5 women are chosen

The Complement of an Event

The Complement of an Event The complement of an event E is the set of outcomes in the sample space that is not in E. We denote the complement of E by E¢. This is a very useful result, since it is often difficult to calculate the probability of an event E but easy to find the probability of E¢.

Example 5 – Finding a Probability Using the Complement of an Event An urn contains 10 red balls and 15 blue balls. Six balls are drawn at random from the urn. What is the probability that at least one ball is red? Solution: Let E be the event that at least one red ball is drawn. It is tedious to count all the possible ways in which one or more of the balls drawn are red. So let’s consider E¢, the complement of this event—namely, that none of the balls that are chosen is red.

Example 5 – Solution cont’d The number of ways of choosing 6 blue balls from the 15 blue balls is C(15,6); the number of ways of choosing 6 balls from the 25 balls is C(25,6). Thus

Example 5 – Solution cont’d By the formula for the complement of an event we have

Events joined with the word The Union of Events Events joined with the word OR

The Union of Events

Example 6 – Finding the Probability of the Union of Events What is the probability that a card drawn at random from a standard 52-card deck is either a face card or a spade?

Example 7 – Finding the Probability of the Union of Mutually Exclusive Events What is the probability that a card drawn at random from a standard 52-card deck is either a seven or a face card?

Probability Of the Intersection of Events

Conditional Probability and the Intersection of Events In general, the probability of an event E given that another event F has occurred is expressed by writing P (E | F) = The probability of E given F For example, suppose a die is rolled. Let E be the event of “getting a two” and let F be the event of “getting an even number.” Then P (E | F) = P (The number is two given that the number is even)

Conditional Probability and the Intersection of Events Since we know that the number is even, the possible outcomes are the three numbers 2, 4, and 6. So in this case the probability of a “two” is P (E | F) Figure 3

Example 9 – Finding the Probability of the Intersection of Events Two cards are drawn, without replacement, from a 52-card deck. Find the probability of the following events. (a) The first card drawn is an ace and the second is a king. (b) The first card drawn is an ace and the second is also an ace. Solution: Let E be the event “the first card is an ace,” and let F be the event “the second card is a king.”

Example 9 – Solution cont’d (a) We are asked to find the probability of E and F, that is, P (E Ç F). Now, P((E) After an ace is drawn, 51 cards remain in the deck; of these, 4 are kings, so P (F | E) By the above formula we have P (E Ç F) = P (E)P (F | E) (b) Let E be the event “the first card is an ace,” and let H be the event “the second card is an ace.” The probability that the first card drawn is an ace is P (E) = After an ace is drawn, 51 cards remain; of these, 3 are aces, so P (H | E) =

Example 9 – Solution By the previous formula we have cont’d By the previous formula we have P (E Ç F) = P (E)P (F | E)

Conditional Probability and the Intersection of Events When the occurrence of one event does not affect the probability of the occurrence of another event, we say that the events are independent. This means that the events E and F are independent if P (E |F) = P (E) and P (F | E) = P (F). For instance, if a fair coin is tossed, the probability of showing heads on the second toss is regardless of what was obtained on the first toss. So any two tosses of a coin are independent.

Example 10 – Finding the Probability of Independent Events A jar contains five red balls and four black balls. A ball is drawn at random from the jar and then replaced; then another ball is picked. What is the probability that both balls are red? Solution: Let E be the event “the first ball drawn is red,” and let F be the event “the second ball drawn is red.” Since we replace the first ball before drawing the second, the events E and F are independent.

Example 10 – Solution cont’d Now, the probability that the first ball is red is The probability that the second is red is also Thus the probability that both balls are red is P (E Ç F) = P (E)P (F)