11 ALGEBRA

EXERCISES 5.2

1.An urn contains 5 red, 3 yellow and 6 white marbles. We draw a marble from the urn at random. What is the probability of drawing a red or a yellow marble?

2.An urn contains 3 black, 4 red and 2 blue marbles. What is the probability that a marble drawn at random is not blue?

3.A fair die is rolled. What is the probability of rolling an even number or a prime number?

4.A fair die is rolled. What is the probability of rolling a number which is less than 5 or greater than 2?

5.You draw a card from a well-shuffled deck of 52 cards. What is the probability of drawing a king or a queen?

6.Two dice are rolled. What is the probability of obtaining a sum of 6 or 10?

7.A bag contains 4 red balls, 3 yellow balls, 5 green balls and 2 black balls. Find the probability that a ball drawn at random is neither black nor red.

b.the spinner will stop on a

green number or an even number?

9.A number is drawn at random from the set {1, 2, 3, …, 150}. Find the probability of drawing a number which is divisible by 3 or 5.

10.A number is drawn at random from the set {2, 4, 6, 8, …, 100}. Find the probability of drawing a number which is not divisible by 3.

11.A traffic light is red for 25 seconds, amber for 5 seconds and green for 40 seconds. What is the probability of arriving at the traffic light when it is not red?

12.A survey of a group of 100 students shows that 70 students own a computer, 40 students own an MP3 player and 20 students own both. Find the probability that a student chosen at random from the group owns a computer or an MP3 player.

13.The face cards in a pack of playing cards are the jacks, queens and kings. A card is chosen at random from a well-shuffled deck of 52 playing cards. Find the probability that it is a black card or a face card.

14. A number is selected at random from the set

A = {1, 2, 3, ..., 180}. What is the probability that the selected number is neither divisible by 3 nor divisible by 5?

15. A group of 7 people is selected at random. What

is the probability that at least two of them were born in the same month?

EXAMPLE 20

Solution

In our probability studies up to now, we have considered sample spaces with only a small number of outcomes. These outcomes can be listed easily. But sometimes the sample space of an experiment has a large number of outcomes. Determining this number might not always be easy or practical. In such cases we can use the counting methods we learned in Chapter 1 to determine the number of outcomes in a sample space and an event.

A number is selected at random from the three-digit numbers formed using the digits {1, 2, 3, 4, 5, 6, 7} without repetition. Find the probability that the selected number is an even number.

The number of outcomes in the sample space is 7 6 5 = 210. There are also three even numbers in the set {1, 2, 3, 4, 5, 6, 7}. So there are three possible choices for the units digit. There are then six possible choices for the hundreds digit and five possible choices for the tens digit.

So the number of outcomes in the event is 6 5 3 = 90 and the desired probability is 21090 = 73.

EXAMPLE 21

Solution

A machine generates all the three-letter sequences of the letters in the word KAHVE, with repetition allowed. Each sequence is written on a card and the cards are put in a box. Ömer draws a card. Find the probability that he draws a sequence beginning with the letter H.

There are 5 5 5 = 125 possible sequences, so n(S) = 125. If H is the event that the

sequence begins with H, n(H) = 1 5 5. So the probability is P(H) n(H) 25 1. n(S) 125 5

EXAMPLE 22

Solution 1

Solution 2

Six people including Murat and Saim are to be seated around a circular table. Find the probability that Murat and Saim are seated next to each other.

We can use the formulas for circular permutation. The number of outcomes in the desired event is (5 – 1)! 2! = 4! 2! and the number of outcomes in the sample space is (6 – 1)! = 5!.

places for him. But only two of the chairs are next to Murat, so the probability that he sits in one of these chairs is 25 .

Solution 1

Solution 2

The number of outcomes in the sample space is n(S) = 28 = 256. The number of outcomes in the desired event is n(E) = C(8, 5) C(3, 3).

Alternatively, we can think of the desired event as an arrangement of the letters in

EXERCISES 5.3

1.Ahmet, Berk, Can, Deniz and Engin are seated at random around a circular table. What is the probability that Ahmet and Can are seated next to each other?

2.Two integers are randomly selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} without repetition. What is the probability that the sum of the numbers is 10?

3.Five math books and 4 physics books are arranged randomly on a bookshelf. What is the probability that books about the same subject are all together?

4.An integer is randomly selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and then replaced. Then another integer is selected. What is the probability that the sum of the two integers is 10?

5.Nine people want to be seated around a circular table. What is the probability that two particular people will not be seated next to each other?

6.Two numbers are drawn at random from 5 positive and 3 negative numbers. What is the probability that the product of the selected numbers is positive?

7.Two numbers are selected randomly from 6 odd numbers and 5 even numbers. What is the probability that the product of the selected numbers is even?

8.In a lottery game, a player must pick 4 winning numbers from 42 numbers to win the prize. Find the probability of winning the prize.

9.5 boys and 4 girls are seated at random in a row of 9 seats. What is the probability that the boys and girls are seated alternately?

10.Two dice are rolled together 900 times. How many times would you expect to get a sum of 7?

11.There are 4 different letters and 6 different mailboxes. Each letter will be put in a random mailbox. What is the probability that all four letters will be put in the same mailbox?

12.200 patients were treated with a new medicine. 32 of them were not cured. Four patients from the 200 are selected at random. What is the probability that three of them were cured?

13.The first 5 positive integers are written randomly in a row. Find the probability that the numbers are written in either ascending or descending order.

14.A four-digit number is formed using the digits {1, 2, 3, 4, 5, 6, 7} without repeating the digits. Find the probability that the number is an even number.

15.A box contains 4 red marbles and 8 other marbles of different colors. What is the probability that 4 marbles selected at random from the box are not red?

16.Seven points A, ..., G are arranged in a circle. A triangle is drawn by connecting three points chosen at random. What is the probability that point E is a vertex of the triangle?