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CHAPTER 7: PROBABILITY •
July 2009
Probability is a way to describe the possibility of something happening.
For example, when the weather report says that there is a 60% chance of rain today, that is an expression of probability. And if someone says that you have a 50 − 50 chance of guessing a coin toss - that too, is an expression e xpression of probability. •
In general, we say that the probability of something happening is the ratio of the number of ways that thing can happen to the total number of ways for all things to happen. The thing we want to happen is usually called the event. So we will need to know the number of ways for the event to happen and the total number of ways for all events to happen. In a simpler form, P ( E ) =
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number of ways for something to happen total no of ways
Probabilities can only take on values from 0 to 1. Keep in mind that 0 and 1 are acceptable values for a probability answer. a nswer. Mathematically this is represented as 0 ≤ P(event) ≤ 1
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A probability of 0 means that an event is impossible and a probability of 1 means that an event is certain. For example, for a dice, the probability of rolling a 7 is zero because you can never roll a 7 with just one dice. P(7) = 0
Probability of mutually exclusive events or Addition Rule •
The addition rule helps you solve probability problems that involve two events.
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Two events are mutually exclusive if they cannot occur at the same time. For example, in an experiment where a dice is thrown, the event of obtaining an even number and the event of obtaining an odd number cannot occur at the same time.
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An event A and an event B are said to be mutually exclusive when: P ( A ∪ B ) = P( A) + P( B) or A ∩ B = O
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For events not mutually exclusive: P ( A ∪ B ) = P( A) + P( B ) − P ( A ∩ B)
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∪
means ‘or’.
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Probability of independent events or Multiplication Rule •
Two events are independent if the outcomes of event A do not influence the outcomes of event B. For example, suppose we roll one dice followed by another and want to find the probability of rolling a 4 on the first die and rolling an even number on the second dice; these two events are independent because their outcomes are not influenced by each other.
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If events A and B are independent, then P( A ∩ B ) = P( A) × P( B)
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∩
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The concepts of the probability of two independent events can be expanded to three or more independents events.
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A tree diagram can be constructed to show all the possible outcomes of an experiment. Each branch shows the possible outcomes of an event.
means ‘and’.
Examples
1.
A letter is randomly selected from the word ‘ LIMAU ’. LIMAU ’. (a) Write down the sample space of this experiment. (b) Determine the number of possible outcomes of the event that the selected letter is (i) a consonant (ii) a vowel.
2.
A bag contains 5 red balls, 8 White balls, 4 green balls and 7 black balls. A ball is drawn at random from the bag. bag. Fine the probability that it is (i) black (ii) not green.
3.
There are 25 red marbles and 20 blue marbles in a box. m blue marbles are then added to the box so that the probability of obtaining a blue marble from the box is 4 . Calculate the value of m. [Ans: 80] 5
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4.
A number is chosen at random from the sample space, S
= {2,4,6,8,10,12,14} .
the probability of obtaining a number less than 9 or greater than 11. [Ans:
5.
]
4 11
]
Determine whether the following event are mutually exclusive or not. •
7.
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In a college, 140 students wear spectacles. If a student is selected randomly, the 1 probability that the student wears spectacles is . Later, another 20 students who 3 wear spectacles join the college. If a student is selected randomly, state the probability that the student wear spectacles. [Ans:
6.
6
Find
A dice is tossed once. (i) getting an even number and getting getti ng an odd number, (ii) getting a prime number and getting gett ing an even number.
8 cards with number from 1 to 8 are placed in a box. A card is picked randomly from the box. (a) Is the events of picking an even number and a prime number mutually exclusive? (b) Find the probability that the card chosen is (i) an even number or a prime number, (ii) an even number and a prime number. 7 1 [Ans: (b)(i)
8
(ii)
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]
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8.
A fair coin and a fair dice are thrown simultaneously. Calculate the probability of obtaining the head of the coin or the number 2 on the dice. [Ans
9.
12
]
Bag A contains four balls numbered 2, 3, 4 and 5. Bag B contains three balls numbered 3, 4 and 6. A ball is drawn at random from each bag. Calculate the probability that both balls have the same number or the sum of the numbers on the two balls is more than 8. [Ans:
10.
7
1 2
]
Two dice are rolled. Find the probability that the sum of the numbers on the uppermost face is (a) an even number or a perfect square. (b) a perfect square or a number less than 5. [Ans: (a)
11 18
(b)
1 4
]
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11.
The probabilities Chong passes his Mathematics, Biology, History test are 0.9, 0.8 and 0.8 respectively. Find the probability that Chong passes all the three subjects. [Ans: 0.576]
12.
Two marbles are drawn at random from the box which contains 5 black marbles, 3 yellow marbles and 2 white marbles. Find the probability that the yellow and black marbles are obtained. [Ans:
13.
6
]
A fair dice and a fair coin are tossed once each. Find the probability that (a) an even number on the dice and a heads on the coin is obtained, (b) a multiple of three and a tails is obtained. [Ans: (a)
14.
1
1 4
(b)
1 6
]
Given that the probability that the soldier hitting the target in each shot is 0.9. If the soldier fires three shots in succession, find the probability that the soldier hits the target twice. [Ans: 0.243]
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15.
Alan, Chai and Daud take an exam and the probabilities that they pass are 3 4 (a) (b) (c)
respectively. Calculate the probability that only one of them passes the exam, at least two of them pass the exam, at least one of them passes the exam.
[Ans: (a)
16.
1 2 , and 2 3
1 4
(b)
17 24
(c)
23 24
]
There are 50 students in a class, of whom 10 are left-handed. Two students are selected at random. What is the probability that (a) both are right-handed, (b) both are left-handed, (c) one is left-handed and the other is right-handed. [Ans: (a)
156 245
(b)
9 245
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(c)
16 49
]
