Topic 1: Probability and Sample Space

I can determine the probability of an outcome in a situation. I can explain, using examples, how decisions may be based on probability. I can solve a contextual problem that involves probability. I can determine if two events are complementary, and explain the reasoning. I can solve a contextual problem that involves the probability of complementary events.

Explore… 1. What is the probability of rolling a 6- sided die and getting: a) a 5 b) not a 5 c) 2 or 3 or 4 (1, 2, 3, 4, or 6)

Explore… 2. If you roll a die 12 times, how many times would you expect the number 4 to show up? 3. Conduct your own experiment by rolling a die 12 times and recording the results 4. In your experiment, what was the experimental proability of rolling a 4. Did 4’s show up the amount of times you expected? Why do you think this is? You would expect that if you roll a die 12 times, tails would show up twice (once every 6 rolls). When you do the experiment, you will likely find this is not the case. This is because the experiment only has 10 trials. If you were to flip the coin 100 or 1000 times, your coin would land on tails closer to half the time. The greater the number of trials, the closer the experimental probability will be to the theoretical probability.

Information Sample space is an organized listing of all possible outcomes from an experiment. Three common ways to organize a sample space include listing the elements, using an outcome table, and drawing a tree diagram.  An event is a collection of outcomes that satisfy a specific condition. For example, when rolling a regular die, the event “roll an odd number” is a collection of the outcomes 1, 3, and 5.

Information Theoretical probability is the mathematical chance of an event taking place. It is equal to the ratio of the number of favorable outcomes to the total number of outcomes. Experimental probability is the ratio of the number of times the event occurs to the total number of trials The complement of any event A, is the event that event A does not occur. We also studied this in Set Theory.

Example 1 Theoretical probability vs. experimental probability There are nine sticker shapes in a container at the local fair. If a player selects the shaded star, he or she wins the game.

Example 1 Theoretical probability vs. experimental probability a) Calculate the theoretical probability of selecting a shaded star on 1 attempt. b) Calculate the theoretical probability of selecting an un-shaded star on 1 attempt.

Example 1 Theoretical probability vs. experimental probability c) Calculate the experimental probability of selecting a start if a blind-folded contestant selects a star six times out of twenty attempts. d) Calculate the theoretical probability of selecting a star shape on one attempt. e) Are the experimental and theoretical probabilities close to equal? (0.3) (0.33333…) They are close to equal.

Example 2 Determining the sample space Determine the sample space of tossing one coin and rolling one die by: a) listing the elements b) drawing a tree diagram {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} H T 1 2 3 4 5 6

Example 2 Determining the sample space c) creating a table d) which method do you like best? e) use the fundamental counting principle to calculate the number of elements in the sample space. 1 2 3 4 5 6 H H1 H2 H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6 The Fundamental Counting Principle can be used to find the number of elements in the sample space by multiplying together the number of outcomes for each event. 6 × 2 = 12

Example 3 Determining the probability A spinner is divided into 3 equal sections as shown. a) If the spinner is spun once i. what is the probability of a loss? ii. What is P(loss’)? Describe what P(loss’) means in words. P(loss’) is a the probability of ‘not a loss.’ In this case, this is the same thing as P(win).

Example 3 b) If the spinner is spun twice, i. create the sample space. Determining the probability W1 L b) If the spinner is spun twice, i. create the sample space.

Example 3 W2 Determining the probability W1 ii. find P(2 wins). iii. find P(at least 1 win). iv. find P(a win and then a loss). v. find P(a win and a loss). Does the order of the win or loss matter? Explain. L This means 1 win or 2 wins. This happens in 8 of the 9 possibilities. This means 1 win and 1 loss. The order can be WL or LW since we are only being asked for a W and a L to occur together.

Need to Know Sample space is an organized listing of all possible outcomes from an experiment. Three common ways to organize a sample space are: listing the elements using an outcome table drawing a tree diagram An event is a collection of outcomes that satisfy a specific condition. For example, when throwing a regular die, the event “throw an odd number” is a collection of the outcomes1, 3, and 5.

Need to Know The number of outcomes in a sample space can be calculated in two ways: by drawing a sample space and counting the number of outcomes by using the Fundamental Counting Principle Theoretical probability is the mathematical chance of an event taking place. It is equal to the ratio of favorable outcomes to the total number of outcomes. Experimental probability is the ratio of the number of times the event occurs to the total number of trials.

Need to Know The probability of any event is represented by a fraction or decimal between 0 and 1. It can also be expressed as a percent when indicated. The sum of all probabilities in a sample space is 1 or 100%. P(A) = 1 means that the probability of event A occurring is 100%, or guaranteed. P(A) = 0 means that the probability of event A occurring is 0%, or impossible.

You’re ready! Try the homework from this section. Need to Know The complement of any event A, is the event that event A does not occur. It can be written as . You’re ready! Try the homework from this section.