4.1 What is Probability?


 Elinor Snow
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1 4.1 What is Probability? between 0 and 1 to indicate the likelihood of an event. We use event is to occur. 1
2 use three major methods: 1) Intuition 3) Equally Likely Outcomes Intuition  prediction based on previous outcomes. Relative Frequency  we have already discussed what relative frequency is when we looked at Probability Formula for Relative Frequency Probability of an Event = f n Wherefis the frequency of an event, andnis the sample size. Example: What is the probability of selecting a female student in this class? In the long run, as the sample size increases and increases, the relative frequency of outcomes get closer and closer to the theoretical (or actual) probability value. An example of how the law of large numbers works is gambling at a casino. Equally Likely Outcomes  when events have the same chance of happening. Example: The probability of correctly guessing the answer to truefalse questions. Probability of an Event = number of outcomes favorable to an event total number of outcomes Can you think of any other situations where there are equally likely outcomes? 2
3 Sample Space results in a definite outcome. Usually the outcome is in the form of a description, count, or measurement. For example: If you toss a coin, there are only 2 possible outcomes (heads or tails). Sample Space set of all possible outcomes. It is especially convenient to know the sample space where all outcomes are likely because then we can compute probabilities of various events using the following formula. total number of outcomes 3
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5 Five Important Facts about Probability probability is, the more likely the event is. 3) The sum of the probabilities of outcomes in a sample space is 1. 4) Probabilities can be assigned by using three methods: intuition, relative or the formula for equally likely outcomes. 5) The probability that an event occurs plus the probability that the same event does not occur is 1. 5
6 Examples 1) If the probability that an event will occur is p, what is the probability that the event will not occur? a) p c) p  1 d) 1  p 2) If the probability that an event will occur is x/4, what is the probability that the event will not occur? a) (1  x)/4 c) (4  x)/x d) (4  x)/4 3) If two fair dice are tossed once, the probability of getting 12 is 1/36. What is the probability of not getting 12? a) 35/36 b) 6/36 c) 30/36 d) 34/36 4) On a test the probability of getting the correct answer to a certain question is represented by x/7. Which of the following can not be a value of x? a) 1 c) 7 5) When a number is chosen at random from the set {1,2,3,4,5,6}, which one of the following events has the greatest probability of occurring? a) not choosing either 1 or 6 c) choosing a number greater than 3 b) choosing an even number d) choosing a prime number 6) The probability of drawing a red marble from a sack of marbles is 2/5. Which one of the following sets of marbles could the sack contain? a) 4 red marbles and 6 green marbles c) 2 red marbles and 5 green marbles b) 6 red marbles and 15 green marbles d) 2 red marbles, 1 blue marble, 4 white marbles probability that it is not green? b) 1/9 c) 5/9 d) 4/9 probability that it is not green? a) 3/5 b) 2/5 c) 4/5 d) 1/5 9) The footlights of a stage have 12 red bulbs, 8 blue bulbs, and 10 yellow bulbs. If all the bulbs are expected to last the same amount of time, what is the probability that a yellow bulb will burn out first? a) 20/30 b) 10/20 c) 10/30 d) 1/30 10) During a half hour of television programming, eight minutes is used for commercials. If a television set is turned on at a random time during the half hour, what is the probability that a commercial is not being shown? b) 1 c) 22/30 d) 0 6
7 4.2 Some Probability Rules Compound Events Independent Events  events where the occurrence or nonoccurrence of one does not For example: deck. Since you replaced it, the probabilities when you select the 2nd card do not change. the probabilities of your second selection would change. Dependent Events  when the outcome of the first event changes the probability of the next event. Does the independence or dependence of an event matter? Independence or dependence determines the way we compute probability of two For Independent Events For Dependent Events " " " Example: If you want to compute the probability of drawing an ace or a king on 2 It is important to distinguish between the"or" combinations and the"and" combinations because we apply different rules to compute the probabilities. Probability Rules: "OR" Problems  add "AND" Problems  multiply 1) Dice 2) Deck of 52 Cards Examples: 1) The probability of throwing two fours on a single toss of a pair of dice is a) 1/6 c) 1/12 d) 1/36 2) If two coins are tossed the probability of getting two tails is a) 1/8 c) 1/4 3) If two cards are drawn from a standard deck of 52 cards without replacement, what is the probability that both cards are fives? a) 4/52 3/52 b) 5/52 4/51 c) 1/4 1/3 d) 2/52 4) From a deck of 52 cards, two cards are randomly drawn without replacement. What is the probability of drawing two hearts? a) 13/52 12/51 b) 13/52 13/51 c) 2/52 d) 13/52 13/51 5) If two cards are drawn from a standard deck of 52 cards without replacement, what is the probability that both cards will be black aces? a) 2/52 2/51 b) 4/52 3/51 c) 4/52 4/51 d) 2/52 1/51 6) If 2 cards are dealt randomly from a standard deck of 52 cards, what is the probability that they are both red queens? a) 2/52 1/51 b) 2/26 c) 4/52 31/51 d) 2/52 7) From a standard deck of 52 cards, two cards are drawn at random without replacement. What is the probability that both cards drawn are aces? a) 12/2,652 b) 4/2,652 c) 4/52 d) 6/2,652 probability of obtaining, at random and without replacement, two yellow gumballs? b) 36/121 c) 30/121 are chosen at random without replacement, what is the probability that both marbles will be yellow? a) 3/14 b) 7/56 c) 1/3 drawn at random, what is the probability both are blue? a) 6/9 b) 30/72 c) 2/9 d) 30/81 7
8 If events are dependent, the occurrence of one event changes the probability of the The notation P(A B) is read the probability P(A, given B) equals the probability that occurred. 8
9 Examples 9
10 Combination of Several Events The addition rule for mutually exclusive events can be extended so that it applies to the situation in which we have more than two events that are mutually exclusive to all other events. 10
11 Summary 11
12 12
13 2. Multiplication Rule of Counting 3. Permutations 4. Combinations 13
14 a method of listing outcomes of an experiment consisting of a series of activities 14
15 15
16 possible outcomes for event E possible outcomes for the series of events E possible outcomes for event E then there are a total of This rule extends to outcomes created by a series of three, four, or more events. Just simply multiply the number of events to get the total number of outcomes for the series. Factorial Notation For any counting number ( ),! = ( 1)( 2) 1 0! = 1 1! = 1 16
17 Examples Evaluate 4! Evaluate 5! Evaluate 6! Evaluate 7! Jean is making sandwiches for a class picnic. She is using 4 different fillings with 2 different kinds of bread. How many different kinds of sandwiches can she make using one kind of filling on one kind of bread for each sandwich? On a restaurant menu, there are six sandwich choices and three beverage choices. How many different lunches may a person order consisting of one sandwich and one beverage? John has 6 pairs of pants and 3 shirts. How many possible outfits consisting of one shirt and one pair of pants can he select? alternative CD's, 1 country CD, 2 Jazz CD's, and 2 Pop CD's from which to choose. How many different combinations of CD's can be played? stripes, and all the other items are solid colors. If Josh will not wear stripes and checkered patterns together, how many different shirt and pants combinations can Josh wear? 17
18 Permutations: Permutation Permutations are especially useful when the order of the data is important. Therefore, we need to enter the data in this order: n n 18
19 Examples 1. Evaluate 7 P 3 2. Evaluate 4 P 3 3. Evaluate 9 P 2 4. Evaluate 8 P 4 5. Evaluate 10 P 3 6. How many different 4letter arrangements can be formed using the letters of the word JUMP, if each letter is used only once? 7. How many different fivedigit numbers can be formed from the digits 1,2,3,4, and 5 if each digit is used only once? 8. How many different 6letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once? 9. All sevendigit telephone numbers in a town begin with 245. How many telephone numbers may be assigned in the town if the last four digits do not begin or end in a zero? 19
20 Combinations: Combinations In combination problems, order is not taken into consideration. Therefore, the difference between permutations and combinations is that in permutations we are considering groupings and in combinations we are considering only the number of 20
21 Examples 1. Evaluate 7 C 3 2. Evaluate 9 C 3 3. Evaluate 10 C 2 4. Evaluate 8 C 6 5. Evaluate 4 C 3 6. Find the number of combinations of 6 things taken 3 at a time. 7. How many different committees of 3 people can be chosen from a group of 9 people? 8. positions individual play while making the first selection, how many teams can be formed if 14 candidates try out and the coach selects 5 players? 9. of 22 songs can be selected? 21
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