The probability that there will be at least 12 successes in the total 18 trials with 60% rate of success is ¹⁸C₁₂(0.6)¹²(0.4)⁶+¹⁸C₁₃(0.6)¹³(0.4)⁵+¹⁸C₁₄(0.6)¹⁴(0.4)⁴+ ¹⁸C₁₅(0.6)¹⁵(0.4)³ +¹⁸C₁₆(0.6)¹⁶(0.4)²+ ¹⁸C₁₇(0.6)¹⁷(0.4)¹+ ¹⁸C₁₈(0.6)¹⁸(0.4)⁰
Binomial Probability distribution, In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a series of n independent experiments.
We have given that :
Rate of sucess (p) = 60% = 0.6
Rate of failure (q) = 1 - 0.6 = 0.4
Number of trials (n) = 18
We want to find P(there will be at least 12 sucess)
Use binomial distribution to find given mentioned probability.
X ~ Bin( 18, 0.6)
P (X= x ) = ⁿCₓ(p)ˣ(q)⁽ⁿ⁻ˣ⁾
Let, X= Number of success among 18
So that it suitable for binomial distribution with parameters are n and p.
P(there will be atleast 12 success)
= P(X>= 12)
18
= ∑ ⁿCₓ (p)ˣ(q)⁽ⁿ⁻ˣ⁾ =
x = 12
¹⁸C₁₂(0.6)¹²(0.4)⁶+¹⁸C₁₃(0.6)¹³(0.4)⁵+¹⁸C₁₄(0.6)¹⁴(0.4)⁴+ ¹⁸C₁₅(0.6)¹⁵(0.4)³ +¹⁸C₁₆(0.6)¹⁶(0.4)²+ ¹⁸C₁₇(0.6)¹⁷(0.4)¹+ ¹⁸C₁₈(0.6)¹⁸(0.4)⁰
= say x
Hence, the required probability is x
To learn more about binomial Probability distribution, refer:
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