Consider an unbiased cubic dice with opposite faces coloured identically and each face coloured red, blue or green such that each colour appears only two times on the dice. If the dice is thrown thrice, the probability of obtaining red colour on top face of the dice at least twice is _______
Concept:
Binomial distribution
\(P\left( {x = k} \right) = {n_{{C_k}}}{p^k}{q^{n - k}}\)
where
p = Probability of success in one trial
q = Probability of failure in one trial = 1 – p
n = Total number of independent trials
k = Discrete random variable
Calculation:
n = 3
\(P\left( {each\;colour} \right) = \frac{2}{6} = \frac{1}{3}\)
\(q = 1 - p = 1 - \frac{1}{3} = \frac{2}{3}\)
Using binomial distribution
Probability of getting red on the top face at least twice is
P(x ≥ 2) = P(x = 2) + P(x = 3)
\(P\left( {x \ge 2} \right) = {n_{{C_2}}}{p^2}{q^{n - 2}} + {n_{{C_3}}}{p^3}{q^{n - 3}}\)
\(\begin{array}{l} P\left( {x \ge 2} \right) = {3_{{C_2}}}{\left( {\frac{1}{3}} \right)^2}{\left( {\frac{2}{3}} \right)^1} + {3_{{C_3}}}{\left( {\frac{1}{3}} \right)^3}{\left( {\frac{2}{3}} \right)^0}\\ = \frac{6}{{27}} + \frac{1}{{27}} = \frac{7}{{27}}=0.259 \end{array}\)
Important Questions from Discrete Distributions
The value of a and b so that the following is probability mass function
X: 0 1 2 P(X = x): 3a 3b 4b with mean 1.1, is:
Digital data received from a sensor can fill up 0 to 32 buffers. Let the sample space be
S = {0, 1, 2, .........., 32} where the sample j denote that j of the buffers are full and \(p\left( i \right) = \frac{1}{{561}}\left( {33 - i} \right)\)
. Let A denote the event that the even number of buffers are full. Then p(A) is :If X is a Poisson random variate with mean 3, then P(|X- 3| < 1) will be:
Let x ∼ N(μ, σ2) If μ2 = σ2, (μ > 0), then the value of P(X < -μ | X < μ) in terms of cumulative function N (0, 1) is:
Consider a binomial random variable X. If X1, X2,...Xn are independent and identically distributed samples from the distribution of X with sum \(Y = \mathop \sum \limits_{i = 1}^n {X_i}\) then the distribution of Y as n → ∞ can be approximated as.
