Probability Review

A committee of 6 members is chosen from a group of 12 Republicans & 10 Democrats. What is the probability that there will be no more than 2 Democrats?

(12C4 / 22C6) + (12C5 / 22C6) + (12C6 / 22C6)

A box contains 4 red cubes & 7 white cubes. If 3 cubes are drawn, what is the probability that they are the same color?

(4C3 / 11C3) + (7C3 / 11C3) = 13/55

How many permutations can be made when the number formed from digits 1, 2, 1, & 3 must be a multiple of 3?

1 + 2 + 1 + 3 = 7; 7 isn't divisible by 3. Therefore, 0.

10 slips of paper numbered from 6 to 15 are placed in a hat. 1 slip of paper is drawn. What is the probability that the number on the slip drawn is prime?

1/10 * 3 = 3/10

What is the probability of getting a number less than 4 in a toss of die?

1/3 * 3 = 1/2

What is the probability of drawing a red 7 from a bridge deck?

1/52 * 2 = 1/26

What is the probability of getting a 2 or 5 in a toss of die?

1/6 * 2 = 1/3

What is the probability of getting a multiple of 3 in a toss of die?

1/6 * 2 = 1/3

How many lines are determined by 10 points, no 3 of which are on the same line?

10 - 1 = 9, 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45

10 members of a football team huddle in a circle around the quarterback to hear the next play. In how many ways can they arrange themselves?

10! / 10 = 362,880

How may ways can you choose a committee of 5 girls & 7 boys from a group of 10 girls & 11 boys? (Find the number of combinations for girls, then for boys. Then find the product.)

10C5 * 11C7 = 83,160

A committee of 6 members is chosen from a group of 12 Republicans & 10 Democrats. What is the probability that there will be no Republicans?

10C6 / 22C6 = 210/74,613

How many ways can 6 ice cream flavors be selected out of 11 flavors?

11C6 = 462

How many ways can 3 identical math books, 4 identical physics books, & 7 identical English books be placed on a shelf?

3 + 4 + 7 = 14, 14! / 3! 4! 7! = 120,120

How many ways can 5 boys be selected for a basketball team out of a physical education class of 30 boys?

30C5 = 142,509

How many permutations can be made when the number formed from digits 1, 2, 1, & 3 must be an even number?

4 - 1 = 3, 3! = 6

A card is drawn from a bridge deck of 52 cards. Then, without replacing the 1st card, a 2nd card is drawn. Find the probability that the 1st card is a queen & the 2nd card is a spade.

4/52 * 26/51 = 2/51

How many ways can you draw 2 cards from a deck of 52 cards?

52C2 = 1,326

How many ways can you draw 56 marbles from a bag that contains 59 marbles?

59C56 = 32,509

In how many ways can 6 people be lined up for a photograph if Ralph must stand immediately to the left of Bernice?

6 - 1 = 5, 5! = 120

How many permutations can be made from the digits in the numeral 121221?

6! / 3! 3! = 20

A bag contains 3 red cubes & 6 white cubes. What is the probability of choosing a cube that isn't red?

6/9 = 2/3

At the head table at a banquet are seated 2 senators, 2 governors, & 3 mayors. Find the number of ways in which these 7 people can be seated with a governor at each end.

7 - 2 = 5, 5! / 2! 3! = 10

Each of 7 cheerleaders stands in a line holding a pompom. There are 3 white pompoms, 2 blue ones, & the rest are yellow. In how many distinguishable ways can the pompoms be arranged?

7! / 3! 2! 2! = 210

How many ways can you choose a group of 4 or more from a group of 7?

7C4 + 7C5 + 7C6 + 7C7 = 64

How many permutations of the letter in the word REPLICA end in a vowel?

7P3 = 210

How many 4-digit integers can be formed from the digits 0, 1, 2, 3, 4, 5, & 6 when no digit is repeated in an integer?

7P4 = 840

How many permutations of 4 letters each can be formed from the letters of the word NUMBERS?

7P4 = 840

How many 4-digit integers can be formed from the digits 0, 1, 2, 3, 4, 5, & 6 when repetitions are allowed?

7^4 = 2,401

In how many ways can 6 out of 8 people be seated in 6 empty seats?

8P6 = 20,160

Find the number of batting orders for 9 baseball players when the best 2 batters must be in the second & third positions.

9 - 2 = 7, 7! * 2 = 10,080

Find the number of permutations that can be made from the letters of the word SATELLITE.

9! / 2! 2! 2! = 45,360

At a safety program, a traffic instructor wishes to display 9 traffic signs. In how many different ways can she display the signs in a row when there are 4 identical stop signs, 2 identical direction signs, & 3 identical speed limit signs?

9! / 4! 2! 3! = 1,260

In how many ways can 9 different books be arranged on a shelf?

9! = 362,880

Odds

Number of successes / number of failures

Probability

Number of successful outcomes / number of possible outcomes

What is the probability AND odds of drawing a 3 in a random selection of the numbers 1, 2, 3, 4, 5, 6, 7, & 8?

Probability = 1/8; odds = 1/7

What is the probability AND odds of drawing an even number in a random selection of the numbers 1, 2, 3, 4, 5, 6, 7, & 8?

Probability = 4/8 = 1/2; odds = 4/4 = 1