# Probability in mathematics

## Introduction to probability

The probability of something happening is the likelihood or chance of it happening. If p is the probability of an event happening and q is the probability of the same event not happening, then the total probability is p + q and is equal to unity, since it is an absolute certainty that the event either does or does not occur, i.e. p + q = 1 Or means + in probability

## Expectation

The expectation, E, of an event happening is defined in general terms as the product of the probability p of an event happening and the number of attempts made, n, i.e. E = pn.
Thus, since the probability of obtaining a 3 upwards when rolling a fair dice is 1/6 , the expectation of getting a 3 upwards on four throws of the dice is 1/6 × 4, i.e. 2/3
Thus expectation is the average occurrence of an event.

## Types of events in probability

1. Dependent event
A dependent eventis one in which the probability of an event happening affects the probability of another event happening. Let 5 transistors be taken at random from a batch of 100 transistors for test purposes, and the probability of there being a defective transistor, p1, be determined. At some later time, let another 5 transistors be taken at random from the 95 remaining transistors in the batch and the probability of there being a defective transistor, p2, be determined. The value of p2 is different from p1 since batch size has effectively altered from 100 to 95, i.e. probability p2is dependent on probability p1. Since 5 transistors are drawn, and then another 5 transistors drawn without replacing the first 5, the second random selection is said to be without replacement.
2. Independent event: An independent event is one in which the probability of an event happening does not affect the probability of another event happening. If 5 transistors are taken at random from a batch of transistors and the probability of a defective transistor p1 is determined and the process is repeated after the original 5 have been replaced in the batch to give p2, then p1 is equal to p2. Since the 5 transistors are replaced between draws, the second selection is said to be with replacement

## Laws of probability

1. The addition law of probability
The addition law of probability is recognized by theword ‘or’joining the probabilities. If pA is the probability of event A happening and pB is the probability of event B happening, the probability of event A or event B happening is given by pA+ pB (provided events A and B are mutually exclusive, i.e. A and B are events which cannot occur together).
Therefore, or means +(plus) in probability
2. Multiplication law of probability: The multiplication law of probability is recognized by the word ‘and’ joining the probabilities. If pA is the probability of event A happening and pB is the probability of event B happening, the probability of event A and event B happening is given by pA× pB. Therefore, and means ×( multiply) in probability
3. Conditional probability: Conditional probability is concerned with the probability of say event B occurring, given that event A has already taken place. If A and B are independent events, then the fact that event A has already occurred will not affect the probability of event B. If A and B are dependent events, then event A having occurred will effect the probability ofevent

### Problem 1. Determine the probabilities of selecting at random (a) a man, and (b) a woman from a crowd containing 20 men and 33 women.

Solution
(a) The probability of selecting at random a man,
p, is given by the ratio
number of men/number in crowd
i.e.p = 20/(20 + 33)
= 20/53 or 0.3774

(b) The probability of selecting at random a women
q, is given by the ratio number of women/number in crowd
i.e.q =33/(20 + 33) = 33/53 or 0.6226

(Check: the total probability should be equal to 1;
p = 20/53 and q = 33/53,
thus the total probability, p + q = 20/53+33/53
= 1 (hence no obvious error has been made)

### Problem 2. Find the expectation of obtaining a 4 upwards with 3 throws of a fair dice.

Solution
Expectation is the average occurrence of an event,and is defined as the probability times the number of attempts. The probability, p, of obtaining a 4 upwards for one throw of the dice is 1/6 .
Also, 3 attempts are made, hence n = 3 and the expectation, E, is pn, i.e. E = 1/6 × 3 = 1/2 or 0.50

### Problem 3. Calculate the probabilities of selecting at random:(a) the winning horse in a race in which 10 horses are running,(b) the winning horses in both the first and second races if there are 10 horses in eachrace.

(a) Since only one of the ten horses can win, the probability of selecting at random the winning horse is;
number of winners/number of horses, i.e.1/10 or 0.10

(b) The probability of selecting the winning horsein the first race is 1/10
. The probability of selecting the winning horse in the second race is 1/10
The probability of selecting the winning horses in the first and second race is given by the multiplication law of probability, i.e.
probability; 1/10 × 1/10 =1/100 or 0.01

1. In a batch of 45 lamps there are 10 faulty lamps. If one lamp is drawn at random, find the probability of it being (a) faulty and
(b) satisfactory
2. A box of fuses are all of the same shape and size and comprises 23 2A fuses, 47 5A fuses and 69 13A fuses. Determine the probability of selecting at random (a) a 2A fuse, (b) a 5A fuse and (c) a 13A fuse.

University question
There are 8 green balls, 4 blue balls and 3 white balls in a box. Then 1 green and 1 blue balls are taken from the box and put away. W hat is the probability that a blue ball is selected at random from the box?
A. 3/13
B. 2/13
C. 4/15
D.3/15

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