Study Notes on Probability | Biometry

The below mentioned article provides a study note on probability.

The measurement of relative chance of occurrence of an event from among a set of alternatives can be defined as probability. When in an experiment there are chances of occurrence of many events then the question of probability arises. Such as, if a coin is tossed then either ‘head’ or ‘tail’ will happen; if a dice is thrown then there are possibility of getting 1 or 2 or 3 … or 6.

Probability is a number expressed in a quantitative scale. When one event will not occur at all then the probability of that event is 0, and if there is any event which will happen positively without fail then the probability of that event is 1. But in biological science, mostly we find the probability of any event lies between impossibility to certainty i.e., the value ranges from 0 to 1.

Mathematically probability can be explained in the following way:

If an event can happen in ‘a’ number of ways, and fails to happen in ‘b’ number of ways, then the probability of its happening ‘p’ is written as.

So, if the probability of happening any event is 0.7, then the probability of not happening of that event is 0.3.

Events:

The results of any experiment in all possible forms are said to be events of that experiment. Such as, throwing of a dice has 6 possible outcomes, either 1 or 2 or 3 or 4 or 5 or 6. All these six outcomes are called events of that single experiment.

Null Event:

When there is no chance of getting an event is called null or impossible event. It is symbolically denoted by ɸ. Such as, survival of any human being forever is an impossible event or null event.

Sure Event:

If the likelihood of occurrence of any event is sure then the event is called sure event. Such as, the death of a human being is a sure event.

Equally likely events:

If the likelihood of the occurrence of every event in an experiment is same then those are called as equally likely events. Such as, when a dice is thrown, there is no biasness, there are the possibilities of coming any number 1 to 6 in equal probability, so the events are equally likely events.

Mutually Exclusive Events:

If in an experiment the occurrence of one event prevents or rules out the happening of all other events, then these are called as mutually exclusive events. Such as, when a coin is tossed either ‘head’ or ‘tail’ will come.

The occurrence of one event affects the occurrence of another event, both events cannot occur together, i.e., occurrence of ‘head’ rules out getting ‘tail’ in the same trial. Here the events are connected by the words ‘either’ or ‘or’.

Simple Event and Compound Event:

Any event having only one sample point of a sample space is called simple event and if any event is decomposable into a number of simple events then it is called as compound event.

Such as, if a bag contains 4 white and 6 red balls, and if one ball is drawn then it is simple event, but if two balls are drawn together then the events will be — ‘both the balls are white’, ‘both the balls are red’, ‘one ball is white and another ball is red’ — these are compound events.

The compound events may be of two types:

Independent Event:

Two or more events are said to be independent events when the outcome of one event does not affect or is not affected by the other events. For example, if a coin is tossed twice, the result of second tossing would in no way be affected by the result of first tossing, so these are independent events.

Dependent Event:

The occurrence or non-occurrence of one event in one trail affects the probabilities of other events in other trails are called dependent events. For example, the probability of drawing a queen from a pack of 52 cards is 4/52, but if the card drawn for the first time (queen) is not replaced then the probability of second drawing of a queen is 3/51, as the pack now contains 3 queens and 51 cards.

Addition and Multiplication Rules:

Probability is estimated usually on the basis of following two rules of chances:

1. Addition rule

2. Multiplication rule

Addition Rule:

This rule is applied when two events are mutually exclusive, i.e., both events cannot occur simultaneously. The birth of a male child excludes the birth of a female child in the same trial. The probability that one of several mutually exclusive events will occur is the sum of the probabilities of the individual events.

Mathematically, when two events A and B are mutually exclusive, the chance of occurrence or probability of occurrence of either A or B can be calculated from the following formula:

p (A or B) = p(A) + p(B)

This rule is applicable to any number of mutually exclusive events as follows:

p(E₁ or E₂ or E₃ … E_n) = p(E₁) + p(E₂) + p(E₃) + … + p(E_n)

Example 1:

If a dice is rolled, what is the probability of getting either 3 or 5?

Probability of getting 3 is p(3) = 1/6

Probability of getting 5 is p(5) = 1/6

.^.. Probability of getting either 3 or 5 is p(3) + p(5) = 1/6 + 1/6 = 1/3

Example 2:

What is the probability of getting a king or a joker from a pack of 54 cards?

Probability of getting a king is p(K) = 4/54 = 2/27

Probability of getting a joker is P(J) = 2/54 = 1/27

So, the probability of either a king or a joker is

p(K or J) = p(K) + p(J) = 2/27 +1/27 = 3/27 = 1/9

Addition rule changes when the events are not mutually exclusive, i.e., if two events A and B can occur simultaneously in few cases, then the rule becomes modified in the following way:

p(A or B) = p(A) + p(B) – p(A and B)

Example 3:

What is the probability of getting a king or club from a pack of 52 cards?

In this example, getting a king and a club are not mutually exclusive events as there will be one king which is king of club. So the chance or probability of getting that event should be subtracted.

p(King or Club) = p(King) + p(Club) – p(King and Club)

Probability of King = p(King) = 4/52 = 1/13

Probability of Club = p(Club) = 13/52 = 1/4

Probability of King and Club = p (King and Club) = 1/52

So, p(King or Club) = (1/13 + 1/4) – 1/52 = 4/13

Multiplication Rule:

(a) When the Events are Independent:

Probability of two or more independent events occurring together is the product of the probabilities of individual events.

Symbolically, if p(A) and p(B) are the probabilities of two respective events A and B, and the happening of these two events are independent then the prob­ability of happening both the events together can be calculated with the fol­lowing formula:

p(A and B) = p(A) x p(B)

Thus the rule may be extended to any number of independent events like E₁, E₂, E₃… E_n, and the formula will be as follows:

p(E₁ and E₂ and E₃ … and E_n) = p(E₁) x p(E₂) x p(E₃) x … x p(E_n)

Example 4:

If two dice are thrown simultaneously what is the probability of getting 3 in both the dice?

The probability of getting 3 in first dice is p(A) = 1/6

The probability of getting 3 in 2nd dice is p(B) = 1/6

So, the probability of getting 3 in both the dice is

p(A and B) = p(A) × p (B)= 1/6 × 1/6 = 1/36

(b) When the Events are Dependent:

When the probability of happening one event is affected by the occurrence of another event then it is called condi­tional probability. Such as, conditional probability of happening A, when B has already happened, is denoted as p(A/B); conditional probability of B, and A has already happened, is denoted as p(B/A).

When the two events A and B are occurring simultaneously but any one event has conditional probability then the multiplication rule will be written as:

p(ab) = p(A)p(B/A) or p(B)-p(A/B)

where p(A/B) = Conditional probability of A given that B has happened

p(B/A) = Conditional probability of B given that A has happened

Example 5:

Four cards are drawn consecutively four times from a pack of 52 cards. Find the chances of drawing an ace, a king, a queen and a jack. The cards are not replaced after each withdrawal.

Probability of drawing an ace = p(A) = 4/52

Probability of drawing a king = p(K) = 4/51

Probability of drawing a queen = p(Q) = 4/50

Probability of drawing a jack = p(J) = 4/49

So, the combined probability

p(A and K and Q and J) = p(A) x p(K) x p(Q) x p(J)

= 4/52 x 4/51 x 4/40 x 4/49 = 0.317

Example 6:

Four cards are drawn in four consecutive drawals from a pack of 52 cards without replacing the cards after each drawal. What is the probability of drawing a king in each drawal?

The probability of getting a king in 1st drawal = 4/52

The probability of getting a king in 2nd drawal = 3/51

The probability of getting a king in 3rd drawal = 2/50

The probability of getting a king in 4th drawal = 1/49

So, the combined probability of getting a king in 4 consecutive drawals is

4/52 x 3/51 x 2/50 x 1/49 = 1/270725

Binomial Distribution:

The binomial distribution is one of the most widely used probability distributions of random discrete variates. This process is one where an experiment can result in only one or two mutually exclusive outcome such as success or failure, male or female, dead or alive, etc.

For example, in case of birth of a child there are two possible happenings, the male or female. So, the probability of male child is p = -y and also the probability of female child is q = 1/2.

If the two deliveries of two ladies are considered then there are four possible out­comes.

We can get this probability distribution through the binomial expansion.

(p + q)² = p² + 2pq + q²

In case of three births, we may get 3 males, 3 females, 2 males one female, one male two females, etc., which we can get through expansion of (p + q)³ = p³ + 3p²q + 3pq² + q³.

Therefore for ‘n’ number of events, the expected result may be expressed as (p + q)ⁿ

Significance in Genetics:

An understanding of the laws of probability is of great importance in genetics. Because it helps in:

(i) Forecasting the chance of obtaining certain results from a cross,

(ii) Elucidating the operation of genetic principles, and

(iii) Assessment of goodness of fit of phenotypic ratio in relation to particular genetic principles.

The principles of probability can be applied to obtain the expected phenotypic ratios from multiple hybrid crosses by avoiding the use of complicated checker boards. How­ever, the actual ratios of offspring obtained in a cross, very rarely, exactly tally with the expected results calculated by the principles of probability.

Within certain limit this devia­tion may be attributed due to chance; but when the actual results deviate to a great extent, the factor other than chance is responsible. Considering the variability in biological mate­rials, a probability of 0.05 is as significant.