Chambers

The decision-making committee consists of three people. For the commission's decision to be valid, at least two members must vote similarly. 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 can make the right decision with a probability of 0.86. However, the third committee member 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 answer:

p =  0.9902

Step-by-step explanation:

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

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