Binomial probability

What is the binominal probability that at least 4 of the 6 trials (n=6) are succeeded where φ = 0.50?

Correct answer:

p =  0.3438

Step-by-step explanation:

$$\begin{gathered} \phi =0.50 \\ {p}_4=\left(\genfrac{}{}{0px}{}{6}{4}\right)\cdot {\phi }^4\cdot (1-\phi {)}^{6-4}=\left(\genfrac{}{}{0px}{}{6}{4}\right)\cdot 0.5^4\cdot (1-0.5{)}^{6-4}={\displaystyle \frac{15}{64}}\doteq 0.2344 \\ {p}_5=\left(\genfrac{}{}{0px}{}{6}{5}\right)\cdot {\phi }^5\cdot (1-\phi {)}^{6-5}=\left(\genfrac{}{}{0px}{}{6}{5}\right)\cdot 0.5^5\cdot (1-0.5{)}^{6-5}={\displaystyle \frac{3}{32}}\doteq 0.0938 \\ {p}_6=\left(\genfrac{}{}{0px}{}{6}{6}\right)\cdot {\phi }^6\cdot (1-\phi {)}^{6-6}=\left(\genfrac{}{}{0px}{}{6}{6}\right)\cdot 0.5^6\cdot (1-0.5{)}^{6-6}={\displaystyle \frac{1}{64}}\doteq 0.0156 \\ p=p_4+p_5+p_6=0.2344+0.0938+0.0156=0.3438 \end{gathered}$$

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