For n Bernoulli trials:
X |
0 |
1 |
2 |
... |
n |
P(X) |
1^{st} term of (q + p)^{n} |
2^{nd} term of (q + p)^{n} |
3^{rd} term of (q + p)^{n} |
... |
(n + 1)^{th} term of (q + p)^{n} |
X |
0 |
1 |
2 |
... |
n |
P(X) |
^{n}C_{0}q^{n-0}p^{0} |
^{n}C_{1}q^{n-1}p^{1} |
^{n}C_{2}q^{n-2}p^{2} |
... |
^{n}C_{n}q^{n-n}p^{n} |
This probability distribution of a random variable, x, is called a binomial distribution.
Thus, the general formula for the probability of a given number of successes is given byâ€¦
P(X = x) = P(x) = ^{n}C_{x}q^{n-x}p^{x}
= n!/(x!(n-x)!) q^{n-x}p^{x}, where x = 0,1,2,....,n and q = 1 - p
P (x) is known as the probability function of the binomial distribution.
Ex:
Find the probability of getting two heads, if a coin is tossed six times.
Sol:
Let the event of getting a head on tossing a coin be a success, and the event of getting a tail be a failure.
Let X denote the number of heads obtained, when a coin is tossed six times.
Let p be the probability of getting a head and q be the probability of getting a tail, if a coin is tossed once.
âˆ´ p = 1/2 and q = 1/2
Number of trials, n = 6
âˆ´ X has a binomial distribution with n =6 and p = 1/2
We have P(X = x) = ^{n}C_{x}q^{n-x}p^{x }, where x = 0,1,2,3,....,n
The probability of getting 2 heads = P(X = 2)
= ^{6}C_{2}(1/2)^{6-2}(1/2)^{2}
= ^{6}C_{2}(1/2)^{6}
= 6!/(2!4!) (1/2)^{6}
= (6x5)/(2x1) . (1/2)^{6}
= 15/64
Hence, the probability of getting 2 heads, when a coin is tossed six times = 15/64
For n Bernoulli trials:
X |
0 |
1 |
2 |
... |
n |
P(X) |
1^{st} term of (q + p)^{n} |
2^{nd} term of (q + p)^{n} |
3^{rd} term of (q + p)^{n} |
... |
(n + 1)^{th} term of (q + p)^{n} |
X |
0 |
1 |
2 |
... |
n |
P(X) |
^{n}C_{0}q^{n-0}p^{0} |
^{n}C_{1}q^{n-1}p^{1} |
^{n}C_{2}q^{n-2}p^{2} |
... |
^{n}C_{n}q^{n-n}p^{n} |
This probability distribution of a random variable, x, is called a binomial distribution.
Thus, the general formula for the probability of a given number of successes is given byâ€¦
P(X = x) = P(x) = ^{n}C_{x}q^{n-x}p^{x}
= n!/(x!(n-x)!) q^{n-x}p^{x}, where x = 0,1,2,....,n and q = 1 - p
P (x) is known as the probability function of the binomial distribution.
Ex:
Find the probability of getting two heads, if a coin is tossed six times.
Sol:
Let the event of getting a head on tossing a coin be a success, and the event of getting a tail be a failure.
Let X denote the number of heads obtained, when a coin is tossed six times.
Let p be the probability of getting a head and q be the probability of getting a tail, if a coin is tossed once.
âˆ´ p = 1/2 and q = 1/2
Number of trials, n = 6
âˆ´ X has a binomial distribution with n =6 and p = 1/2
We have P(X = x) = ^{n}C_{x}q^{n-x}p^{x }, where x = 0,1,2,3,....,n
The probability of getting 2 heads = P(X = 2)
= ^{6}C_{2}(1/2)^{6-2}(1/2)^{2}
= ^{6}C_{2}(1/2)^{6}
= 6!/(2!4!) (1/2)^{6}
= (6x5)/(2x1) . (1/2)^{6}
= 15/64
Hence, the probability of getting 2 heads, when a coin is tossed six times = 15/64