Concept:

Binomial Distribution:

• A Binomial Distribution is an experiment in which only ONE OUT OF TWO outcomes is possible. For example: Head or Tail, Yes or No, 1 or 0, etc.
• If ‘n’ and ‘p’ are the parameters, then ‘n’ denotes the total number of times the experiment is conducted and ‘p’ denotes the probability of the happening of the event.
• The probability of getting exactly ‘k’ successes in ‘n’ independent trials for a Random Variable X is expressed as P(X = k) and is given by the formula:

  P(X = k) = nCk pk (1 - p)n - k

Calculation:

For every 3 trials, the experiment succeeds 2 times and fails 1 time.

∴ Probability of success: p = \(\dfrac23\).

For at least 4 success in 6 trials, n = 6 and k = 4, 5, 6. The required probability is therefore:

\(\rm ^6C_4\left(\dfrac23\right)^4\left(1-\dfrac23\right)^{6-4}+^6C_5\left(\dfrac23\right)^5\left(1-\dfrac23\right)^{6-5}+^6C_6\left(\dfrac23\right)^6\left(1-\dfrac23\right)^{6-6}\)

= \(\rm (15)\left(\dfrac23\right)^4\left(\dfrac13\right)^2+(6)\left(\dfrac23\right)^5\left(\dfrac13\right)^1+(1)\left(\dfrac23\right)^6\left(\dfrac13\right)^0\)

= \(\rm \dfrac{240}{729}+\dfrac{192}{729}+\dfrac{64}{729}\)

= \(\rm \dfrac{496}{729}\).

• ⁿC_r = \(\rm \dfrac {n!}{r!(n-r)!}\).
• n! = 1 × 2 × 3 × ... × n.
• 0! = 1.