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VERY Brief Introduction to Probability

Notation: P(x) is read “the probability of x”

Think of P(x) as a ratio. [pic]

Let’s consider a flip of a coin. What is the probability of flipping a coin and getting heads?

[pic]

Let’s consider a roll of a die. What is the probability of rolling a die and getting a 3?

[pic]

Let’s consider the days of the week.

What’s the probability of getting a day of the week that starts with S?

What is the probability of getting a day of the week that starts with Z?

Can we do any worse than that?

What is the probability of getting a day of week that ends in “y”?

Can we do any better than that?

If we let p represent the probability that an event happens, q will be the probability it does not happen.

If the probability p of guessing right on a multiple-choice question is p = 0.25,

What is q, the probability of guessing wrong?

It will always be true that p + q =1 and q can be computed as q = 1 – p.

Probability Distributions

Requirements for P(x) to be a probability distribution:

[pic]

Formulas: Calculator:

[pic] [pic]

Probability Distributions for the number of Heads

One Coin Flip Two Coin Flips Three Coin Flips Four Coin Flips

x P(x) x P(x) x P(x) x P(x)

0 0 0 0

1 1 1 1

2 2 2

3. 3

4

For Four Coin Flips:

1. What is the probability of getting exactly 1 head?

2. What is the probability of getting at least 2 heads?

3. What is the probability of getting at most 3 heads?

Which of the following describe a probability distribution?

1. [pic]the probability of rolling “x” on a single roll of a die [pic]

|x |1 |2 |3 |

|P(x) |1/6 |2/6 |3/6 |

3. [pic]

|x |1 |2 |3 |4 |

|P(x) |1/1 |1/2 |1/3 |1/4 |

4. [pic]

|x |1 |2 |3 |4 |

|P(x) |1/10 |2/10 |3/10 |4/10 |

5. [pic]

|x |0 |1 |2 |3 |4 |

|P(x) |0.15 |0.25 |0.35 |0.45 |-0.20 |

6. What is the missing probability?

|x |0 |1 |2 |3 |4 |

|P(x) |0.15 |0.25 |0.35 |0.15 | |

Find the mean, variance, and standard deviation for the probability distributions below.

7. 8. 9.

x P(x) x P(x) x P(x)

0 .0625 1 .250 1 .4

1 .250 2 .250 5 .3

2 .375 3 .250 10 .2

3 .250 4 .250 25 .1

4 .0625

Quick Quiz 4

1. Find the range, standard deviation and variance for the sample data:

11 10 8 4 6 7 11 6 11 7

Range=

Standard Deviation =

Variance =

2. Determine the missing probability:

x 0 1 2 3 4

P(x) 0.07 0.20 0.38 ? 0.13

3. Is the following a probability distribution? If no, why not?

x 0 1 2 3

P(x) 0.005 0.435 0.555 0.206

4. Find the mean, variance and standard deviation for the probability distribution:

x 0 1 2 3 4

P(x) 0.05 0.25 0.35 0.25 0.10

5. Use the probability distribution in #4 to answer the following:

P( x = 2)

P(x < 3)

Probability x is at least 1

6. A “pick 3” lotto has a $1000 prize. What is the expected outcome if a ticket costs $2?

(The probability of winning is .001).

Answers:

1. Range = 7, S.D. = 2.5, Variance = 6.3 (answer varies based on rounding)

2. 0.22

3. No, the sum of the probabilities is > 1

4. mean = 2.1, S.D. = 1.044, Variance = 1.09 (answer varies based on rounding)

5. 0.35, 0.65, 0.95

6. -$1

Expected Outcomes

1. Suppose you take out a $32,000 insurance policy and pay $400 per year in premiums. If your probability of surviving the year is 99.5%, what is the insurance company's expected profit?

2. When you play the daily lotto you pay $1 and choose a 3-digit number. If you win $500 when you "hit", what are your expected winnings (or losings)?

3. A landscaping company loses $200 on rainy days and gains $1000 on days when it does not rain. If the probability of rain is 25% what is the expected gain for the day?

4. If you have a 10% probability of gaining $200, a 30% probability of losing $300, and a 60% probability of breaking even, what is your expected outcome?

Expectation in Roulette

[pic]

What is your expected outcome when you bet:

Black

The number 13

The first column

The “corner” 13/14/16/17

The “street” 13/14/15

Binomial Experiments

Must meet the following criteria:

1. Fixed number of trials

2. Independent trials

3. Dichotomous categories (2 classifications of outcomes)

4. Probabilities remain constant for each trial

Formulas:

P(x) = nCx · px · qn-x Remember [pic]

mean [pic]

variance [pic]

standard deviation = [pic]

p = P(success) q = P(failure)

p + q = 1 therefore 1 – p = q

Find the mean, variance, and standard deviation for the binomial distributions below.

1. 10 coin flips, x = number of heads

2. 5 dice rolls, x = number of sixes

3. 10 item multiple choice quiz (a,b,c,d,), x = number correct when guessing

Would it be unusual to get a passing grade (6 or more correct) on a 10-item quiz, as described above, when guessing?

Quick Quiz 5

1. Find the mean, variance and standard deviation for the sample data:

15 8 12 5 19 14 8 6 13

Mean = Variance = Standard Deviation =

2. Find the mean, variance and standard deviation for the frequency table:

Mean =

Variance =

Standard Deviation =

3. Find the mean, variance and standard deviation for the probability distribution:

x 1 2 3 4 5

P(x) 0.16 0.22 0.28 0.20 0.14

Mean = Variance = Standard Deviation =

4. Find the mean, variance and standard deviation for a binomial distribution which has:

n = 10 and p = 0.3

Mean = Variance = Standard Deviation =

Using the result, would it be unusual to get a value for x that is 7 or more?

Answers:

1. Mean = 11.1, Variance = 21.2, SD = 4.6 (answers may vary depending on rounding)

2. Mean = 24.7. Variance = 132.3, SD = 11.5

3. Mean = 2.94, Variance = 1.61, SD = 1.27 (answers may vary depending on rounding)

4. Mean = 3, Variance = 2.1, SD = 1.45 (answers may vary depending on rounding)

Yes. Using 2 SD’s we would expect to see x values between 0.1 and 5.9. Seven is out of this range, so it would be unusual.

Finding Binomial Probabilities

Consider 10 coin flips :

P(exactly 5 heads) =

P(exactly 2 heads) =

P(more than 7 heads) =

Consider a quiz: 5 questions, 25% chance of guessing right

P(none right)

P(three right)

P(at least 3 right)

Consider dice rolls: 5 dice

P(exactly 3 sixes)

P(all sixes)

P(less than 3 sixes)

Consider safety devices: 3 devices, 90% effective

P(all work)

P(none work)

P(at least 1 works)

Review of the Binomial Distribution

The probability of guessing right on a 5-option multiple choice item is 1/5 or 0.20.

If a quiz contains 10 questions:

a. What is the probability of getting five right?

b. What is the probability of getting three right?

c. What is the probability of getting at most 3 right?

d. What is the probability of failing (getting less than 6 right)?

e. What is the probability of getting a C or better (getting at least 7 right)?

f. What is the probability of getting more than half right?

g. Find the mean and standard deviation for this binomial distribution.

h. Would it be unusual to get none right?

i. Would it be unusual to get seven right?

If I Were Studying For a Test

On Chapter 6, These Are the Things I Would

Be Sure To Know

1. Find the mean, range, standard deviation and variance for the sample data:

21 20 18 14 16 17 21 16 21 17

Mean =

Range=

Standard Deviation =

Variance =

2. Determine the missing probability:

x 0 1 2 3 4

P(x) 0.08 0.25 0.32 ? 0.15

3. Is the following a probability distribution? If no, why not?

x 0 1 2 3

P(x) 0.05 0.45 0.55 -0.05

4. Find the mean, variance and standard deviation for the probability distribution:

x 0 1 2 3 4

P(x) 0.18 0.34 0.23 0.21 0.04

Mean =

Standard Deviation =

Variance =

5. Use the probability distribution in #4 to answer the following:

P( x = 2)

P(x < 3)

Probability x is at least 1

6. In a game of chance, you have a 30% chance of winning $5, a 50% chance of

losing $4 and a 20% chance of breaking even. What is your expected profit

or loss?

7. The probability of guessing right on a 5-option multiple choice item is 1/5 or 0.2.

If a quiz contains 10 questions:

a. What is the probability of getting five right?

b. What is the probability of failing (getting less than 6 right)?

c. Find the mean and standard deviation for this binomial distribution.

d. Would it be unusual to get none right?

Answers:

1. Use the “1varstat” command using L1 to determine the answers.

Mean = 18.1, Range = 7, S.D. = 2.5, Variance = 6.3

2. Since the P(x)’s must sum to 1, the missing value must be 0.20.

3. Due to the negative P(x) for x = 3, this is not a probability distribution.

4. Use the “1varstat” command using L1 and L2 to determine the answers.

Mean = 1.6, S.D. = 1.1, Variance = 1.2

5. Use the table in number 4 to add the appropriate P(x)’s

P(x = 2) is 0.23, P(x < 3) is 0.75, P(x at least 1) is 0.82

6. Create a probability distribution and use “1varstat” using L1 and L2 to find the mean of the probability distribution. This is the expected outcome, -0.5 or -$.50.

7. a. binompdf(10, 0.2, 5) = 0.026

b. sum(binompdf(10, 0.2, {5, 4, 3, 2, 1, 0}) = 0.993

c. np = (10)(0.2) = 2, [pic]

d. “Unusual” means more than 2 standard deviations away from the mean. Using the mean and standard deviation in “c” anything between -0.6 and 4.6 would be NOT unusual. Since “0” is in that range, it would not be unusual to get none right.

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Mean

Standard Deviation

Expected Outcome

Calculator

“Exactly”

2nd

Distr

binompdf

n,p,x

“Range of Values”

2nd

list

>math

5:sum

enter

2nd

Distr

binompdf

n,p,{x1,x2,…}

Interval f

0-9 5

10-19 11

20-29 18

30-39 10

40-49 6

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