Binomial Probability Distribution for tossing coin by Sir Tanveer

A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice).
For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.
Binomial distributions must also meet the following three criteria:

The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense—if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%.
Each observation or trial is independent. In other words, none of your trials have an effect on the probability of the next trial.
The binomial distribution formula is:

b(x; n, P) = nCx * Px * (1 – P)^n – x
OR
b(x; n, P) = nCx * Px * (q)^n – x
Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials

The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another.
Step 1: Identify ‘n’ from the given question. Using our sample question, n (the number of randomly selected items) is 5.

Step 2: Identify ‘X’ from the problem. X (the number you are asked to find the probability for) is:-
i) Exactly 3 heads
ii) At least 3 heads
iii)At the most 2 heads .

Step 4: Find p and q. p is the probability of success and q is the probability of failure. Here, p = 0.5. and failure q is also 0.5. Because, if there is no p and q is given in the question, then the p and q automatically becomes 1/2.
Remember:- p+q=1

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