The ABC “Needs” of probability calculations



The ABC “Needs” of probability calculations

A = Arithmetic skill (adding, subtracting, multiplying, and dividing)

B = Business-like organization

C = Counting accuracy

(Knowing the formulas below doesn’t hurt either!!)

Some useful formulas to keep in mind (or in hand)

U – Union (or)

∩ - Intersection (and)

Adding probabilities

General Formula P(A U B) = P(A) + P(B) – P(A ∩ B)

Nonmutually exclusive, or overlapping outcomes

Special-Case Formula P(A U B) = P(A) + P(B)

Mutually exclusive, or nonoverlapping outcomes

Multiplying probabilities (“and”/sequential events)

Nonindependence, conditional probabilities; sampling without replacement

P(A ∩ B) = P(A) * P(B|A)

Independence; sampling with replacement; leads to binomial formula

P(A ∩ B) = P(A) * P(B)

Examples (Note: Deck of cards has 52 cards total)

1. If one card is randomly selected from a “deck” consisting of just four aces: What is the probability that it will be an ace of hearts? 1/4

2. If one card is selected from a standard deck of playing cards:

a) What is the probability it will be a heart? 13/52 = 1/4

b) What is the probability it will be a red card or a picture card? P(R) + P(P) - P(R ∩ P) = 26/52 + 12/52 - (26/52 * 12/52) = 13/26 + 6/26 - (3/26) = 16/26 = 8/13.

c) What is the probability it will be either a king or a queen? P(K) + P(Q) = 4/52 + 4/52 = 8/52 = 2/13.

3. If two cards are randomly selected from a deck of playing cards (without replacement): What is the probability of getting two hearts? P(A) * P(B) = 13/52 * 12/51 = 156/2652 = 13/221.

4. If two cards are randomly selected from a deck consisting of just 12 face cards (without replacement): What is the probability of getting two queens?

P(A) * P(B) = 4/12 * 3/11 = 12/132.

5. What is the answer to question 4 if the first card selected is replaced in the deck before the second card is selected?

P(A) * P(B) = 4/12 * 4/12 = 16/144 = 1/9.

6. If a fair coin is flipped three times what is the probability of getting three heads?

= 1/2 * 1/2 * 1/2 = 1/8.

7. If two cards are randomly selected from a deck consisting of just 12 face cards (without replacement): What is the probability of getting a jack and a queen?

P(A) * P(B) = 4/12 * 4/11 = 16/132.

8. What is the answer to question 7 if the first card selected is replaced in the deck before the second card is selected?

P(A) * P(B) = 4/12 * 4/12 = 16/144 = 1/9.

9. Below are the probabilities of the marital status of women from age 20 to 29. Assuming the probabilities were calculated from a random sample of women, if you were to random sample a 1000 women how many of them would be married?

Marital status Never Married Married Widowed Divorced

Probability .404 .535 .004 .057

=.535*1000 = 535

10. A nutritional researcher is interested in the relationship between broccoli and candy corn. The researcher interviews 200 subjects and finds the following.

|Likes Broccoli |Likes Candy Corn (A) |

| |YES (A) |No(not A) |Total |

|Yes(B) |20 |60 |80 |

|No(not B) |100 |20 |120 |

|Column total |120 |80 |200 |

Answer the following questions.

|What is the P(A)? 120/200 |5. What is the P(A ∩ not B)? 100/200 = 1/2. |

|What is the P(B)? 80/200 |What is the P(A|B)? 20/80 |

| | |

|What is the P(A ∩ B)? 20/200 |What is the P(A|not B)? 100/120 |

|What is the P(A U B)? 120/200 + 80/200 - 20/200 = 180/200 =9/10 | |

| | |

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