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LESSON 4 – Expectation and Expected Value Scenario 1: A fair die is rolled 60 times. If a die is rolled once, calculate P(3). How can we use P(3) to determine the number of times we EXPECT to gain a 3 out of the 60 rolls? In general: If there are n members of a sample, and the probability of that event occurring is p then the expectation of the occurrence of that event is: ______________ Text Practice Exercise 9G.1 - Questions 3, 5, 6 New Concept - Expected Value and its Notation Expectation is a useful tool when you need to make predictions about certain situations. People working in fields involving chance often use this mathematics to weigh up potential gains versus potential loss to make informed choices. When we perform a repeated experiment, probability can help Scenario 2: Consider this scenario: Chad records the number of 3 pointers (xi )shot during the basketball season. The results are shown below. xi 0 1 2 3 4 ni 1 3 5 2 1 n = Pi 1 12 3 12 5 12 2 12 1 12 P a. Explain what ni represents in context of the scenario. b. Explain what Pi represents in context of the scenario. c. What is the value of P(x2) and what does it mean? i i d. Calculate n e. Calculate P . Explain what this tells you in context of the scenario. i . Explain what this tells you. i f. Calculate P(x >3) and explain what it tells you. Scenario 3: Let’s PLAY! In a game of chance a spinner with the numbers, 2, 4 and 6 is spun. When the spinner is spun 3 times, we would expect that each of the scores would be spun once. Could you predict what the expected AVERAGE for those 3 spins would be? Hint: For every 3 spins we would expect to get the number 4 one time, so for ONE throw, we would expect to get the number 4, 1/3 times. For one spin we can determine the expected average by: 246 =4 3 Alternatively we could use the notation from above xi 2 4 6 Pi 1 3 1 3 1 3 P 1 3 1 3 1 3 i Expected value = 2 4 6 4 Expected Value = This can be written as a formula below x i pi Practice: Text Exercise 9G.2 page 283 Questions: 1, 3, 4