Precalculus by Richard Wright

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“Don’t be afraid,” the prophet answered. “Those who are with us are more than those who are with them.” 2 Kings‬ ‭6‬:‭16‬ ‭NIV‬‬

# 10-08 Probability

Summary: In this section, you will:

• Calculate simple probability
• Calculate the probability of compound events
• Calculate the probability of multiple events
• Calculate the complement of an event

SDA NAD Content Standards (2018): PC.5.4, PC.7.1, PC.7.3

It is often important to know how likely something is to occur. For example, what is the likely hood of winning the lottery. Give you a hint, you will very likely lose. A general rule of all gambling is that if you play, you will probably lose. The probability is that the organizer of the gambling or lottery will make money because the players lose. Otherwise why go to the effort to organize the games?

Probability is a number from 0 to 1 to indicate how likely something is to happen. 0 means it will never happen, and 1 means it will always happen. Probability is defined as the ratio of favorable outcomes to total possible outcomes. All the possible outcomes is called the sample space.

###### Probability

Number from 0 to 1 indicating how likely something is to happen

0 = Never happen

1 = Always happen

Probability of event A occurring

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

#### Example 1: Simple Probability

A bag contains 15 US dimes, 3 Canadian dimes, and 2 British ten pence. If a coin is selected at random from the bag, what is the probability of choosing a US dime?

###### Solution

Favorable outcomes are choosing one of the 15 US dimes. Total outcomes are choosing one of the 20 coins.

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

$$P(US dime) = \frac{15}{20} = 0.75$$

##### Try It 1

A clearance bin of movies has 10 good movies, 15 average movies, and 75 bad movies. What is the probability of randomly selecting a good movie?

0.1

#### Example 2: Simple Probability

In the game of Settlers of Catan, resources are gained by the sum of two rolled dice. Maria built a village on a 5. What is the probability of rolling a 5?

###### Solution

Make a list of all the possible outcomes of rolling two dice. The red is the first die and the blue is the second die.

 1 1 2 1 3 1 4 1 5 1 6 1 1 2 2 2 3 2 4 2 5 2 6 2 1 3 2 3 3 3 4 3 5 3 6 3 1 4 2 4 3 4 4 4 5 4 6 4 1 5 2 5 3 5 4 5 5 5 6 5 1 6 2 6 3 6 4 6 5 6 6 6

The 4 shaded outcomes are the favorable outcomes, and there are a total of 36 possible outcomes.

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

$$P(S = 5) = \frac{4}{36} = 0.44$$

##### Try It 2

Two coins are flipped at the same time. What is the probability that one coin is a head and other is a tail?

0.5

## Compound Events

Sometimes you need to know the probability of more complicated situations. A compound event is one event with two acceptable outcomes. In the diagram below, either outcome A or outcome B are acceptable. The overlap area is counted twice; once with A and once with B. So the overlap must be subtracted.

$$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$$

The ∪ is the union symbol which means combining all the elements in set A or set B. The ∩ is the intersection symbol which means the overlap of the two sets or all the elements in set A and set B.

If $$P(A ∩ B) = 0$$, then the events are called mutually exclusive.

#### Example 3: Compound Event

You draw one card from a standard 52-card deck. What is the probability of drawing a spade or a face card?

###### Solution

This is a compound event because it is one event with two acceptable outcomes.

$$P(♠ ∪ 😃) = P(♠) + P(😃) - P(♠ ∩ 😃)$$

$$P(♠ ∪ 😃) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52}$$

$$P(♠ ∪ 😃) = \frac{22}{52} = \frac{11}{26} ≅ 0.42$$

##### Try It 3

A coin is tossed and a die is rolled. What is the probability that the coin is a head and the die is a 4?

$$\frac{2}{3} ≈ 0.67$$

## Multiple Events

Other times, the probability of several events is wanted. In general, to find the probability of multiple events, multiply the probabilities of the individual events together. If the events do not affect each other, then they are independent events. Otherwise, the events are dependent if the events do affect each other.

Independent Events: $$P(A \text{ and } B) = P(A) · P(B)$$

Dependent Events: $$P(A \text{ and } B) = P(A) · P(B|A)$$ where $$P(B|A)$$ is the probability of B given that A already happened

#### Example 4: Multiple Events

Two cards are drawn from standard 52-card deck. What is the probability of drawing a spade and a diamond card if (a) with replacement and (b) and without replacement?

###### Solution
1. With replacement means that the first card is drawn, looked at, then put back in the deck. The deck is then shuffled and the second card is drawn. That makes this question independent.

$$P(♠ \text{ and } ♦) = P(♠) · P(♦)$$

$$P(♠ \text{ and } ♦) = \frac{13}{52} · \frac{13}{52}$$

$$P(♠ \text{ and } ♦) = \frac{1}{4} · \frac{1}{4} = \frac{1}{16} = 0.0625$$

2. Without replacement means that the first card is drawn and held, then the second card is drawn. After the first card is drawn, there are less cards to draw, so the probabilities are now different. That makes this situation dependent.

$$P(♠ \text{ and } ♦) = P(♠) · P(♦|♠)$$

$$P(♠ \text{ and } ♦) = \frac{13}{52} · \frac{13}{51}$$

$$P(♠ \text{ and } ♦) = \frac{1}{4} · \frac{13}{51} = \frac{13}{204} ≈ 0.0637$$

##### Try It 4

A student guesses on a two question T/F quiz. What is the probability that the student gets both questions correct?

0.25

## Complements

A complement of a probability is the probability of the opposite. Sometimes it is much easier to find the probability of the opposite of what you are supposed to find. Key words like "at least" or "at most" indicate that finding a complement might be the easiest way to solve the problem.

$$P(\overline{A}) = 1 - P(A)$$

where $$P(\overline{A})$$ is the complement of P(A)

#### Example 5: Complement

A store has 5 calculators on the rack, but 2 are defective. If you randomly choose two calculators, what is the probability that at least one is defective?

###### Solution

At least one is defective means that 1 is defective or 2 are defective. It is easier to find the opposite which would be 0 defective or both not defective.

$$P(d ≥ 1) = 1 - P(d = 0)$$

The probability of getting 0 defective calculators is the same as getting 1 good calculator followed by another good calculator without replacement.

$$P(d = 0) = P(\text{good})P(\text{good}|\text{good})$$

$$P(d = 0) = \frac{3}{5} · \frac{2}{4} = \frac{3}{10}$$

Now find the probability of the complement.

$$P(d ≥ 1) = 1 - P(d = 0)$$

$$P(d ≥ 1) = 1 - \frac{3}{10} = \frac{7}{10} = 0.7$$

##### Try It 5

Your sibling is late to leave for school about 4 times a week. What is the probability that they will be late at least once in a week?

$$\frac{1}{25} = 0.04$$

##### Lesson Summary

###### Probability

Number from 0 to 1 indicating how likely something is to happen

0 = Never happen

1 = Always happen

###### Simple Probability

Probability of event A occurring

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

###### Compound Events

1 event with 2 accepted outcomes (OR)

$$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$$

###### Multiple Events

2 or more events (AND)

Independent Events: $$P(A \text{ and } B) = P(A) · P(B)$$

Dependent Events: $$P(A \text{ and } B) = P(A) · P(B|A)$$

###### Complement

Opposite

$$P(\overline{A}) = 1 - P(A)$$

## Practice Exercises

1. What is the difference between compound events and multiple events?
2. Three coins are flipped. Calculate the probability. (Hint: First list the sample space.)

3. Exactly 2 heads
4. 3 tails
5. One six-sided die is rolled. Calculate the probability.

6. Even number
7. A multiple of 3
8. One card is drawn from a standard 52-card deck. Calculate the probability.

9. A face card or a 10
10. A red card or an ace
11. Two six-sided dice are rolled. Calculate the probability.

12. Sum is 11
13. Sum is prime or even
14. Two marbles are drawn from 8 red marbles, 5 blue marbles, and 7 yellow marbles in a bag. Calculate the probability of drawing

15. A blue and a yellow (a) with replacement (b) without replacement
16. A red and another red (a) with replacement (b) without replacement
17. Problem Solving

18. Jill makes a free throw 90% of the time. What is the probability that she will miss a free throw?
19. Billy has a 0.23 probability of finishing his homework on time. What is the probability that his homework will be late?
20. The types of birds at my bird feeder in the morning and afternoon are given in the table.

Sparrows Woodpeckers Doves Total
Morning 26 4 2 32
Afternoon 33 5 3 41
Total 59 9 5 73
1. What is the probability that a random bird is a sparrow?
2. What is the probability that a random bird appears in the morning?
3. What is the probability that a random bird is a woodpecker that comes in the afternoon?
21. The percent of people following the major world religions is given the circle graph. Data is taken from Pew Research in 2010 when there were 6.9 billion people in the world.

1. Estimate the number of non-Christians.
2. A person is selected at random. What is the probability that they are a Muslim?
3. A person is selected at random. What is the probability that they are a Buddhist or Hindu?
22. Francine is trying to test out of Spanish class, but she has never learned any Spanish. The test has ten multiple choice questions with five choices each.
1. What is the probability that she will randomly guess all ten questions correctly?
2. What is the probability that she will guess at least one answer correctly?
23. Two cards are drawn from a standard deck of cards. What is the probability of drawing two face cards without replacement?
24. A shipment of 12 calculators contains 4 defective units. What is the probability that a school which purchases 4 calculators will receive (a) 4 good units, (b) 3 good units, and (c) at least 2 good units.
25. In a certain NASA rocket, the guidance system and its backup function 99% of the time.
1. What is the probability of both systems functioning?
2. What is the probability of both systems failing?
3. What is the probability of at least one system functioning?
26. PowerBall is a multistate lottery game. A player chooses 5 white balls from a set of 69 and 1 red ball from a set of 26. The order that the balls are chosen is not important. Find the probability of winning PowerBall.
27. Mixed Review

28. (10-07) How many different orders can six books be arranged on a bookshelf?
29. (10-07) How many different license plates can be made if each one is 4 letters followed by 1 number?
30. (10-06) Expand $$(x - 3y)^3$$.
31. (10-05) Prove $$1 + 2 + 4 + 8 + \cdots + 2^{n-1} = 2^n - 1$$.
32. (10-04) Evaluate $$\displaystyle \sum_{i=1}^{10} 2^{i-1}$$.

1. A compound event is 1 event with 2 accepted outcomes. Multiple events are more than one event.
2. $$\frac{3}{8} = 0.375$$
3. $$\frac{1}{8} = 0.125$$
4. $$\frac{1}{2} = 0.5$$
5. $$\frac{1}{3} ≈ 0.333$$
6. $$\frac{4}{13} ≈ 0.308$$
7. $$\frac{7}{13} ≈ 0.538$$
8. $$\frac{1}{18} ≈ 0.056$$
9. $$\frac{8}{9} ≈ 0.889$$
10. $$\frac{7}{80} ≈ 0.088; \frac{7}{76} ≈ 0.092$$
11. $$\frac{4}{25} = 0.16; \frac{14}{95} ≈ 0.147$$
12. 0.1
13. 0.77
14. 0.808; 0.438; 0.068
15. 4.69 billion; 0.23; 0.22
16. $$≈ 1.02×10^{-7}$$; ≈ 0.893
17. ≈ 0.050
18. 0.141; 0.453; 0.933
19. 0.9801; 0.0001; 0.9999
20. $$\frac{1}{292,201,338} ≈ 3.42×10^{-9}$$
21. 720
22. 4,569,760
23. $$x^3 - 9x^2y + 27xy^2 - 27y^3$$
24. Show work and last step is $$2·2^k - 1$$
25. 1023