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BASIC PROBABILITY

Unit 2 Day 5

Definitions• Experiment – process that gives definite results• Outcomes – results of an experiment• Sample space – set of all possible outcomes

Examples• Experiment – Tossing a coin–Outcomes: Heads or Tails–Sample space: S = {H, T}

• Experiment – Rolling a die–Outcomes: 1, 2, 3, 4, 5, and 6–Sample space: S = {1, 2, 3, 4, 5, 6}

ExampleConsider this dartboard. Assume that the experiment is

“throwing a dart” once and that the dart hits the board. Find each of the following.a) The outcomesb) The sample space

Solution:a) The outcomes are hitting white (W), purple (P), or yellow (Y).b) The sample space is {hitting white, hitting purple, hitting

yellow}, which can be stated as {W, P, Y}.

Definition of an EventIf S is a sample space of an experiment, then an event

is any subset of the sample space.

Examples…1. Die showing an even number2. Picking an ace from a deck of cards

ExampleIf an experiment consists of tossing a coin three times and recording the results in order, find the sample space. (Find all possible outcomes)

{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

ExampleThe event E of showing “exactly two heads” is the

subset of S that consists of all outcomes with two heads. Write out that possible event.

E = {HHT, HTH, THH}

ExampleWhat is the event F of showing “at least two heads”?

F = {HHH, HHT, HTH, THH}

What is the event G of showing “no heads”?

G = {TTT}

Types of Probability• Experimental Probability–Based on trials and observations

• Theoretical Probability–Calculated by analyzing a situation

Experimental ProbabilitySociological Survey. The

authors of this text conducted an experimental survey to determine the number of people who are left-handed, right-handed, or both. The results are shown in the graph.a) Determine the probability that a person is right-handed.b) Determine the probability that a person is left-handed.

Solutions:a) The number of right-handed is 82, the number of left-handed is 17, the number of ambidextrous is 1. The total number of observations is 82 + 17 + 1 = 100.The probability that a person is right-handed is

P = 82/100 = .82 = 82%

b) The probability that a person is left-handed is P, where

P = 17/100 = .17 = 17%

Probability Facts• P(E) is a number between 0 and 1,

0 ≤ P(E) ≤ 1• If an event is certain to occur, then

P(E) = 1• If an event is impossible, then

P(E) = 0• The closer the probability of event is to 1, the more

likely the event is to happen.

Examples• If you flip a coin, what is the theoretical

probability that it lands with heads up?– ½ or 50%

• If you flip a coin, what is the theoretical probability that it lands with tails up?– ½ or 50%

• How would you find experimental probability?

Examples• If you roll a standard die, what is the

theoretical probability that it lands with the 3 facing up?– 1/6

• If you roll a standard die, what is the theoretical probability that it lands with the 3 or the 4 facing up?– 2/6 or 1/3

ExampleSuppose we select, without looking, one marble

from a bag containing 4 red and 9 purple marbles. What is the probability of selecting a red marble?

Solution: There are 13 equally likely ways of selecting any marble, and 4 ways of selecting red.

P(selecting a red marble) = 4/13

Example

What is the probability of getting a sum of 5 on a roll of a pair of dice?

Rolling a Pair of Dice

SolutionOn each die, there are 6 possible outcomes. The outcomes are paired so there are 6(6) or 36 possible ways in which the two can fall.

There are 4 ways to roll a total of 5:(1, 4) (4, 1) (2, 3) and (3, 2).

P(sum of 5) = 4/36 = 1/9

Your Turn!• What is the probability of choosing, at random, the

ace of spades from a deck of 52 cards?• What is the probability of choosing any ace from a

deck of 52 cards?• What is the probability of drawing a red card from a

deck of 52 cards?• What is the probability of drawing a club from a

deck of 52 cards?

Solutions• What is the probability of choosing, at random, the

ace of spades from a deck of 52 cards? 1/52• What is the probability of choosing any ace from a

deck of 52 cards? 4/52 = 1/13• What is the probability of drawing a red card from a

deck of 52 cards? 26/52 = 1/2• What is the probability of drawing a club from a

deck of 52 cards? 13/52 = 1/4

ExampleA five-card poker hand is drawn from a standard deck

of 52 cards. What is the probability that all five cards are spades?

= picking 5 spades picking 5 cards

= 13(12)(11)(10)(9) 52(51)(50)(49)(48)

= 0.000495 or 0.0495 %

Solution• How many different ways can we choose five spades

from 13 spades?

• Probability of drawing five spades is

ExampleA bag contains 20 tennis balls, of which four are

defective. If two balls are selected at random from the bag, what is the probability that both are defective?

= pick a defected tennis ball picking any two tennis balls

Solution

= 4(3) 20(19)

= 0.8316 or 83.16 %

HomeworkProbability WS