/    /  Statistics – Probability

Statistics – Probability:

The probability theory is very helpful for making predictions. In research investigation, estimates and predictions form an important part.  Using statistical methods, we estimate for the further analysis. The role of probability in modern science is simply a substitute for certainty. Probability can be defined in terms of a random process giving rise to an outcome. Rolling a die or flipping a coin is a random process which gives rise to an outcome.

Probability of occurrence of an event A is

P(A) =Number of favorable outcomes/Total number of equally likely outcomes

Choose a random number from 1 to 5. What is the probability of each outcome?

The possible outcomes are 1, 2, 3, 4 and 5.

P(1) = no of ways to choose a 1/total numbers =1/5

Similarly each number can occur p(2)=p(3)=p(4)=p(5) =1/5

An event with probability 0 has no chance of occurring whereas, an event of probability 1 is certain to occur.

Probability always takes values between 0 and 1(inclusively).

It may also be represented as a percentage between 0% and 100%.

As more observations are collected, the proportion n of occurrences with a particular outcome converges to the probability p of that outcome.

Here are some of the rules in finding probabilities in different situations:

Two outcomes are said to be disjoint or mutually exclusive if they both cannot happen.

If two events, A and B, are mutually exclusive, the probability that A or B will occur is:

P (A or B) = P(A) + P(B)

If two events, A and B, are non-mutually exclusive, the probability that A or B will occur is:

P(A or B) = P(A) + P(B) – P(A and B)

A single random card is chosen from a deck of 52 playing cards. What is the probability of choosing a Queen or a club?

Probabilities:

P(queen or club)=P(queen)+P(Club)-P(queen of clubs)

=4/52+13/52-1/52

=16/52

=4/13

If two events A and B are independent, the probability of both occurring is:

P(A and B) = P(A) · P(B)

If a coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 5 on the die.

Probabilities: 

P(head) =1/2

P(5)=1/6

P(head and 5) =P(head ).P(5)

=1/2 . 1/6

=1/12

Conditional probability is P(B|A) read as probability of event B given that event A has already occurred.

P(B|A) =P(A and  B)/P(A)

A math teacher gave her class two tests. 23% of the class passed both tests and 46% of the class passed the first test. What is the percentage of those who passed the first test also passed the second test?

P(Second|First) =P(First and second)/P(First) =0.23/0.46=0.50=50%

If two events, A and B, are dependent, the probability of both occurring is:

P(A and B)  =  P(A) · P(B|A)

Mr. Parth needs two students to help him with a science demonstration for his class of 16 girls and 12 boys. He randomly chooses one student who comes to the front of the room. He then chooses a second student from the remaining class. What is the probability that both chosen students are girls?

P(G1 and G2) = P(G1) and P(G2|G1)

=16/28 .15/27

=240/756

=60/189