Biathlete shooting probability

According to long-term statistics, a biathlete has a shooting success rate of 72%.
a) what is the probability that 4 targets will be hit on one item (5 shots).
b) what is the probability that fewer than 4 targets are hit on one item (5 shots)?

Final Answer:

p₁ =  0.3762
p₂ =  0.4303

Step-by-step explanation:

$$\begin{gathered} p_0=72\%=\frac{72}{100}=\frac{18}{25}=0.72 \\ {p}_1=\left(\genfrac{}{}{0ex}{}{5}{4}\right)\cdot p_0^4\cdot (1-p_0{)}^{5-4}=(\genfrac{}{}{0ex}{}{5}{4})\cdot 0.72^4\cdot (1-0.72{)}^{5-4}=5\cdot 0.72^4\cdot 0.28^1=0.3762 \end{gathered}$$

$$\begin{gathered} p_5=\left(\genfrac{}{}{0ex}{}{5}{5}\right)\cdot p_0^5\cdot (1-p_0{)}^{5-5}=(\genfrac{}{}{0ex}{}{5}{5})\cdot 0.72^5\cdot (1-0.72{)}^{5-5}=1\cdot 0.72^5\cdot 0.28^0\doteq 0.1935 \\ {p}_2=1-(p_1+p_5)=1-(0.3762+0.1935)=0.4303 \end{gathered}$$

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