Chambers

The decision-making committee consists of three people. In order for the commission's decision to be valid, at least two members must vote in the same way. It is not possible not to vote in the commission, everyone only votes yes or no. We assume that the first two members of the commission are experts and each of them can make the right decision with a probability of 0.86. However, the third member of the commission is not an expert, so he decides completely randomly (for example, he throws a coin), so the probability that he will make the right decision is only 0.5. What is the probability that the whole decision-making committee will make the right decision?

Consider that the three-member chambers of the Supreme Court decide similarly . .. .

Correct result:

p =  0.9902

Solution:

$$\begin{gathered} p_1=0.86 \\ {p}_2=p_1=0.86={\displaystyle \frac{43}{50}} \\ {p}_3=0.5 \\ P(A\cup B\cup C)=P(A)+P(B)+P(C)\text{−}P(A\cap B)\text{−}P(A\cap C)\text{−}P(B\cap C)+P(A\cap B\cap C) \\ p=p_1+p_2+p_3-p_1\cdot p_2-p_1\cdot p_3-p_2\cdot p_3+p_1\cdot p_2\cdot p_3=0.86+0.86+0.5-0.86\cdot 0.86-0.86\cdot 0.5-0.86\cdot 0.5+0.86\cdot 0.86\cdot 0.5=0.9902 \end{gathered}$$

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