Bell Work: Collect like terms: x + y – 1 – x + y + 1

Answer: 2y

LESSON 32: PROBABILITY

Probability*: the likelihood that a particular event will occur. To represent the probability of event A we use the notation P(A). We express probability as a number ranging from zero to one.

Range of Probability Impossible Unlikely Likely Certain Probability is a ratio of favorable outcomes to possible outcomes. P(Event) = Number of favorable outcomes Number of possible outcomes 0½1

Example: The spinner show at the right is spun once. What is the probability the spinner will stop a) In Sector A? b) In Sector A or B? c) In Sector A, B, C, or D? AB CD

Answer: a) ¼ b) ½ c) 4/4 or 1

Probability may be expressed as a fraction, as a decimal, or as a percent. We often use the word chance when expressing probability in percent form. If something has a ¼ chance of being picked, then it also has a 25% chance.

Odds is the ratio of the number of favorable to unfavorable outcomes and is often expressed with a colon.

Example: If the spinner in the previous example is spun once, what are the odds the spinner will stop in Sector A?

Answer: of the four equally likely outcomes, one is A and three are not A. So the odds the spinner will stop in sector A are 1 : 3.

The sample space of an experiment is the collection of all possible outcomes. We can record the sample space in a variety of ways, including a list or a table.

For example, if a coin is tossed twice, there are four possible outcomes. In the list and table we use H for “heads” and T for “tails”. Sample Space = {HH, HT, TH, TT} Sample Space Second Toss First Toss HT HHHTH THTTT

In the previous example, what is the probability of getting heads at least once in two tosses of a coin?

Answer: ¾

Complement*: the set of outcomes in the sample space that are not included in the event.

The complement of getting heads at least once in two tosses is not getting heads at least once. The probability of an event and the probability of its complement total one, because it is certain that an even either will occur or will not occur. If we know the probability of an event, we can find the probability of its complement by subtracting the probability from 1.

Example: The spinner in example 1 is spun twice. a) predict: is the probability of spinning A more likely with one spin or with two spins?

Answer: We guess that we will be more likely to get A with two spins than with one spin.

Example: b) Find the probability of getting A at least once.

Answer: Seven of the 16 outcomes Result in A at Least once, so the probability of A is 7/16. 1 st Spin 2 nd Spin ABCD AAABACADA BABBBCBDB CACBCCCDC DADBDCDDD

Example: c) Find the probability of not getting A at least once.

Answer: The probability of not A is the complement of the probability of A. 1 – 7/16 = 9/16

Example: Nathan flips a coin three times. Predict which is more likely, that he will get heads at least once or that he will not get heads at least once.

Answer: The probability of getting heads in one toss is ½. Since the opportunity to get heads at least once increases with more tosses, we predict that getting heads at least once is more likely than not getting heads.

Example: Find the probability of getting heads at least once.

Answer: Since 7 of the 8 outcomes have heads at least once, P(H) = 7/8. H T First Toss H T H T < < < < Second Toss H T H T H T H T Third TossOutcomes HHH HHT HTH HTT THH THT TTH TTT

Example: Find the probability of not getting heads at least once.

Answer: 1 – 7/8 = 1/8 The probability of not getting heads is 1/8.

We distinguish between the theoretical probability, which is found by analyzing a situation, and experimental probability, which is determined statistically.

Experimental probability is the ratio of the number of times an event occurs to the number of trials. For example, if a basketball player makes 80 free throws in 100 attempts, then the experimental probability of the player making a free throw is 80/100 or 4/5. the player might be described as an 80% free-throw shooter.

In baseball and softball, a player’s batting average is the experimental probability, expressed as a decimal, of the player getting a hit.

Example: Emily is a softball player who has 21 hits in 60 at-bats. Express the probability that Emily will get a hit in her next at-bat as a decimal number with three decimal places. Emily’s coach needs to decide whether to have Emily bat or to use a substitute hitter whose batting average is.300 (batting averages are usually expressed without a 0 in the ones place). How would you advise the coach?

Answer: The ratio of hits to at-bats is 21 to 60. to express the ratio as a decimal we divide 21 by ÷ 60 = 0.35 We write the probability (batting average) with three decimal places:.350. since Emily’s batting average is greater than the batting average of the substitute hitter, we advise the coach to have Emily bat.

HW: Lesson 32 #1-30 Due Tomorrow