MAT 209 - Statistics Chapter 4 - Possibilities and Probabilities

Chapter 4 Possibilities and Probabilities

Probability = The art of “how to live with uncertainties”

4.1 Counting

Two ways to answer a “what is possible” question:

·      Listing everything that can happen in a given situation (tree diagrams).

·      Determines how many things can happen w/o creating a list.
Multiplication of choices: If a choice consists of two steps, of which the first can be made in m ways and for each of these the second can be made in n ways, then the whole choice can be made in m*n ways.
Generalizations of Multiplication of choices: If a choice consists of k steps with the 1^st step made in n ways, for each of these the 2^nd step can be made in n₂ ways, and for each combination of choices made in the first two steps can be made in n₃ ways … with the final step made in n_k ways, then the whole choice can be made in n₁*n₂*n₃ … n_k ways.

4.2 Permutations

Permutation = If r objects are selected from a set of n distinct objects, in any particular arrangement.

Number of permutations of r objects selected from a set of n Distinct objects:

Or in factorial notation,

Number of permutations of n distinct objects taken altogether:

4.3 Combinations

Combination = If r objects can be selected from a set of n distinct objects, but we do NOT care about the order in which the selection is made.

The number of ways in which r objects can be selected from a set of n distinct objects:

Or in factorial notation,

For  we often replace with  referred to as a binomial coefficient. So .

This formula can be extended to rule for binomial coefficient:

4.4 Probability

The Classical Probability Concept: If there are n equally likely possibilities, of which one must occur and s are regarded as favorable, or as success, then the probability of a success is s/n.

Application of classical probability concept with combination:

Example 4.21 Find the probability that 2 cards drawn from an ordinary deck of 52 playing cards will both be black.

Total number of possibilities is

Total number of favorable possibilities is

Based on these numbers the possibility of a favorable =

Example 4.22 A manufacturer of cell phones plans to ship 12 cell phones and as a precaution test checks 4 of the phones. The 4 phones checked were found to be satisfactory. However, the 12 cell phone shipment had 2 defectives. What is the probability that the 4 phones checked will contain no defects, when actually 2 of the 12 phones are defective?

Possible ways of choosing 4 of 12 cell phones:  = 495 ways.

Possible ways of choosing 4 of 10 good phones is selected:  ways.

Probability is 210/495 = 0.42424242… ~ 42%

Major shortcoming of classical probability is limited applicability because many situations where various possibilities cannot be regarded as equally likely.

For repeat events:

Frequency Interpretation: The probability of an event (happening/outcome) is the proportion of the time that events of the same kind will occur in the long run. à Cannot guarantee event, but can mathematically compare the proportion of success.

Law of Large Numbers: If a situation, trial or experiment is repeated, again and again, the proportion of successes will tend to approach the probability that any one outcome will be a success.

For single events:

·      Comparison to other “similar” events for data. à Risk of different conditions that validate “similar events.”

·      Evaluating the accuracy of the methods used to predict the outcome.

·      Probabilities interpreted as personal or subjective evaluations.

4.5 Mathematical Expectation

Mathematical expectation = Product of the amount a player stands to win and the probability that he/she will win. If the probability of obtaining amounts a₁, a₂, …, or a_k are respectively p₁, p₂, …, p_k, then the mathematical expectation is

Example 4.27 What is the mathematical expectation if we buy 1 of 2000 raffle tickets issued for a 1^st prize TV ($540), 2^nd prize tape recorder ($180), and 3^rd prize radio ($40)?

1,997 raffle tickets will have 0 payoffs, 1 will have $540, 1 will have $180, 1 will have $40.

Total merchandise prize $760. à .

Probability of loss:

Probability of prize:

On average:

Equitable or fair game = Game that does not favor either player; each player’s mathematical expectation is zero.

Decision Problem

For decisions problems, it is best to select the most promising mathematical expectation.

Payoff tables help keep multiple factors organized and easy to read.

Example 4.32 Manufacturer must decide whether or not to invest in new factory. Knows that if new factory is built with goods sales year = $451,000 profit; if new factory built with poor sales = - $110,000 deficit; if factory NOT built with good sales $220,000; if factory NOT built with poor sales = $22,000 profit. Probability for good sales = 0.40, for poor sales = 0.60.

Payoff table	New Factory	NO factory
Good Sales	$451,000	$220,000
Poor Sales	-$110,000	$22,000

If new factory built expected profit

If factory NOT built expected profit 220,000(0.40) + 22,000(0.60) = $101,200.

$114,400 > $101,200 à New factory should be built to maximize potential profits.

Bayesian Analysis : Statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Options are assigned probabilities with varying states of uncertainties (states of nature). Best selection is the alternative that promises the greatest expected profit or smallest expected loss.