CHAPTER 1: PROBABILITY 1. Introduction to Probability L EARNING TARGET: I CAN DETERMINE THE PROBABILITY OF AN EVENT. What s the probability of flipping heads on a coin? Theoretically, it is 1/2 1 way to flip heads out of two equally likely faces that could show. How could you calculate the probability of flipping ten consecutive heads? What about the probability of flipping at least one head on ten flips? In this section you will learn how to determine probabilities for a variety of situations. Probability Probability is defined as the ratio of ways an event can succeed to the total number of possibilities: TA 1.1: Counting Probs 1.2: Combinations Chapter Overview: 1.: Intro to Prob 1.4: Extended Prob 1.5: Geometric Prob 1.6: Expected Value 1.7: Ch. Review # of # of successes possible outcomes Probability is often written as a simplified fraction, but it can also be written as a decimal or percent. Odds Odds is the ratio of successes to failures. ODDS = successes: failures In simple cases, probability and odds can be calculated by counting possible outcomes using a sample space. Example #1: If two dice are rolled, what is the probability of rolling a sum of 7? A sample space can be created showing all possible dice rolls: As you can see from the table, 6 of the 6 possible rolls result in a sum of 7, so the final probability is 6/6 = 1/6. 1
1. INTRODUCTION TO PROBABILITY Compound Probability Probability of Mutually Exclusive Unions (OR Problems) What is the probability of selecting a 5 or a black ace from a standard deck of cards? Since the events are mutually exclusive, there are six ways out of 52 to select a winner four 5 s and two black aces. A second approach would be to add the two probabilities: 4 2 6 5) black ace ) 52 52 52 26 Probability of Event A or B (Mutually Exclusive) Aor Probability of Non-Mutually Exclusive Unions (OR Problems) What is the probability of drawing a black OR an ace from a standard deck? Based on the previous example, you would expect it to be the sum of the probability of a black card 26/52 and the probability of an ace 4/52. However, selecting a black or an ace are not mutually exclusive events there are two black aces (the intersection of the two events). As a result, those two black aces are counted both with the black cards and the aces. To account for that over-count, the probability of the intersection must be subtracted from the sum: 26 4 2 28 black ) ace ) black ace ) 52 52 52 52 7 1 Probability of Event A or B (Non-Mutually Exclusive) Aor A Probability of Intersections (AND Problems): Dependent Events Determine the probability of selecting a King from a standard deck AND then drawing a second king without replacement. Method #1 Counting: There are 4 = 12 different ways to select a king, selected by a second king. There are 52 51 = 2652 ways to select two cards from the deck. So the final probability is 12/2652 = /66. Method #2 Multiply Probabilities: You can also multiply the probability of the first event by the probability of the second event (given that the first has already occurred). This is: 4 King) another king) 52 51 Probability of Dependent Events A and B (No Replacement) Aand B A is the probability of event A given that B has already occurred 2
Probability of Intersections (AND Problems): Independent Events Calculate the probability of flipping heads on four consecutive coins. Coin flips don t affect other coin flips, so the events are independent (so are playing card problems involving replacement). In general, try to write compound probability problems using AND or OR. Flipping four heads in a row is the probability of flipping heads AND heads AND heads AND heads: heads) heads) heads) heads) = (1/2)(1/2)(1/2)(1/2) = 1/16 Probability of the Complement of an Event: What is the probability of flipping at least one head if three coins are flipped? This COULD be solved by breaking the problem into AND/OR scenarios. However, it would take a lot of calculations, because it could be three heads, or two heads, or one head, plus there are several possible orders in which the results could occur: heads in a row) OR H, H, T) OR H, T, H) OR T, H, H), OR T, T, H) OR T, H, T) OR P (H, T, T) It would be very time-consuming to make each of these calculations. However, it would be very easy to find the complement of the event that is, flipping NO HEADS: Probability of the Complement of an Event: 1 - T, T, T) = T) T) T) = (1/2)(1/2)(1/2) = 1/8 If there is a 1/8 probability of flipping NO HEADS, there must be a 1-1/8 = 7/8 probability of the opposite flipping at least one head!! Conditional Probability Conditional probability is the probability of an event occurring given that another event has already taken place. This often involves looking at a subset of data to calculate the probability based on the given circumstances. For example, the probability that a roll of a die is a 4 given that it is known to be even is just 1/, since there are even possibilities, and one is a 4. For more challenging situations, the formula below can be used: Conditional Probability of A Given That B Has Occurred: A A For the above problem, the probability of the intersection of A and B (rolling an even 4) is 1/6, while the probability of event B (rolling an even) is 1/2. This results in (1/6) (1/2) = 1/.
Q Problems Q1.) To the nearest hundredth, find 11.9% of 87.6. Q2.) 40% of a number is 24. What is the number? Q.) Convert to scientific notation: 12,576. Q4.) A car travels 220 miles in 4 hours. What is its velocity? Q5.) Solve for all possible solutions of x: x 5 = 12 Q6.) Write in simplified radical form: Q7.) A parallelogram with a right angle must be classified as a: Q8.) What kind of angles are <1 and <2? Q9.) Two legs of a right triangle measure 10 and 24 inches. How long is the hypotenuse? Q10.) If the side lengths of a triangle are tripled, the area increases how many times? 1. If a fair die is rolled, determine the probability of: Rolling a Rolling a or a 5 C) Rolling an even number D) rolling a prime or an even number 2. Calculate the probability given the odds: :2 2:5 C) 5:1. Calculate the odds given the probability: 8/15 2/ C) 1/5 4. Calculate the following probabilities using two methods: i) By creating a sample space ii) By showing how it can be calculated without using a sample space Flipping three heads in a row Flipping three of the same result in a row C) Flipping at least two heads (on flips) 5. Create a sample space to calculate the following probabilities on a roll of two fair dice: sum of 7) sum of 6) C) sum of 8) D) sum of 5) E) sum of or 6) F) sum of 1) G) sum of at least 10) 6.) Based on your sample space for #5, which is more likely rolling a sum of 7) or rolling a sum of or 4)? 7.) What is the probability of rolling a 2 on a die and then flipping a heads on a coin? 4
8. Carlos is trying to get dressed in the dark in his bedroom. His drawer has 5 black, 4 brown, and white socks. How many socks does he have to draw to guarantee getting a pair of black socks? What is the probability he selects a black or brown sock? C) What is the probability he chooses two brown in a row without replacement? D) Determine the probability that he selects two socks of the same color. (Hint: Break down the problem into smaller parts using AND/OR!) E) Calculate the probability of NOT drawing a white sock on two draws (assume he does not replace the first sock). F) What is the probability he picks five black socks in a row? 9. A cooler of drinks contains 7 Cokes, 4 Sprites, and 6 root beers. Find each probability, assuming no replacement: Coke or Sprite) Coke, then a Sprite) C) Coke & Sprite, either order) D) root beers in a row) E) Drawing three in a row without getting a Sprite) F) At least one Sprite on three draws) (Be crafty!!) 10. The probability that a traffic light will be red is 0%. What is the probability of being stopped at the first light and not at the second? What is the probability of three red lights in a row? C) What is the probability of at least one green light out of three? 11. The probability that it will rain Sunday is 80%. The probability it will rain Monday is 60%. Determine: rains both days) rains Sunday, not Monday) C) rains neither day) D) rains Sunday or Monday) 12. If Joey Votto s batting average is.20 (he gets a hit 2% of the time), what is the probability: He gets two hits in a row He gets out two at-bats in a row C) He gets at least one hit in two at-bats D) He gets EXACTLY one hit 1. When drawing from a standard deck of 52 cards without replacement, what is the probability of choosing: A face card followed by an ace Two consecutive cards of the same color C) Three consecutive cards of the same suit D) Three cards, all of different suits E) A black card or a king F) A heart or a 4 G) A king or a queen followed by a black jack 14.) Last season the Bengals passed on first down 40% of the time (and ran the ball 60% of the time). If they passed on first down, the probability they passed on second down was 0%. If they selected a running play on first down, the probability they passed on second down was 80%. Find the probability the Bengals will: pass on first and second down. pass on first down, but not second down C) pass on second down, but not on first down D) pass on neither first nor second down E) Add the four probabilities. How do you explain this result? 15.) A coin is flipped five times. What is the probability of flipping at least one tails? 16.) If a basketball makes 60% of his free throws, what is probability that he makes EXACTLY 2 out of shots? R1.) The measure of the supplement of an angle is 0º less than five times the measure of the complement. What is the measure of the angle? 5