Statistics & Probability: The Rules of Probability

Probability

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

Probability = \(\frac{number\hspace{0.5mm}of\hspace{0.5mm}favorable\hspace{0.5mm}outcomes}{number\hspace{0.5mm}of\hspace{0.5mm}possible\hspace{0.5mm}outcomes} \)

BE CAREFUL!

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Make sure that you apply the correct formula for the probability of an event.

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Probability is a value between 0 (event does not happen) and 1 (event will happen). For example, the probability of getting heads when a coin is flipped is 12 because heads is 1 option out of 2 possibilities. The probability of rolling an odd number on a six-sided number cube is \(\frac{3}{6} = \frac{1}{2}\) because there are three odd numbers, 1, 3, and 5, out of 6 possible numbers.

The probability of an “or” event happening is the sum of the events happening. For example, the probability of rolling an odd number or a 4 on a six-sided number cube is \(\frac{4}{6}\). The probability of rolling an odd number is \(\frac{3}{6}\), and the probability of rolling a 4 is \(\frac{1}{6}\). Therefore, the probability is \(\frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\).

The probability of an “and” event happening is the product of the probability of two or more events. The probability of rolling 6 three times in a row is  \(\frac{1}{216}\). The probability of a single event is  \(\frac{1}{6}\), and this fraction is multiplied three times to find the probability, \(\frac{1}{6} × \frac{1}{6} × \frac{1}{6}\). There are cases of “with replacement” when the item is returned to the pile and “without replacement” when the item is not returned to the pile.

The probability of a “not” event happening is 1 minus the probability of the event occurring. For example, the probability of not rolling 6 three times in a row is \(1 –  \frac{1}{216} = \frac{215}{216}\).

Examples

Probability Review

Calculating Expected Values and Analyzing Decisions Based on Probability

The expected value of an event is the sum of the products of the probability of an event times the payoff of an event. A good example is calculating the expected value for buying a lottery ticket. There is a one in a hundred million chance that a person would win $50 million. Each ticket costs $2. The expected value is

\(\frac{1}{100,000,000}(50,000,000 – 2) + \frac{ 99,999,999}{100,000,000}(–2) =  \frac{49,999,998}{100,000,000} – \frac{199,999,998}{100,000,000} = – \frac{150,000,000}{100,000,000} = –$1.50\)

On average, one should expect to lose $1.50 each time the game is played. Analyzing the information, the meaning of the data shows that playing the lottery would result in losing money every time.

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BE CAREFUL!

The expected value will not be the same as the actual value unless the probability of winning is 100%.

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Examples

Calculating Expected Values Review

Let’s Review!

• The sample space is the number of outcomes of an event. The union, the intersection, and the complement are related to the sample space.
• The probability of an event is the number of possible events divided by the total number of outcomes. There can be “and,” “or,” and “not” probabilities.
• The expected value of an event is based on the payout and probability of an event occurring.