Probability problem...

[1a2s3d4f5g6h7j8k9l]

[FONT=Greg\]The probability that Anna can solve a certain math problem is .7, that Bobbi can solve the same problem is .5 and that Cate can solve it is .8. Find the probability that at least two of them will solve the math problem.


​Please help...thank you all![/FONT]

 

[soroban]

Hello, 1a2s3d4f5g6h7j8k9l!

The probability that Anna can solve a certain math problem is 0.7,
that Bobbi can solve the same problem is 0.5, and that Cate can solve it is 0.8.
Find the probability that at least two of them will solve the math problem.

We are given: .\(\displaystyle \begin{Bmatrix} P(A) &=& 0.7 && P(\overline{A}) &=& 0.3 \\ P(B) &=& 0.5 && P(\overline{B}) &=& 0.5 \\ P(C) &=& 0.8 && P(\overline{C}) &=& 0.2\end{Bmatrix}\)

We assume their solving abilities are independent,
. . so that, for example: .\(\displaystyle P(A \cap B) \,=\,P(A)\cdot P(B)\)


"At least two" means: "exactly two or all three".

. . \(\displaystyle \begin{array}{ccccc}P(A\cap B \cap\overline{C}) &=& (0.7)(0.5)(0.2) &=& 0.07 \\ P(A \cap\overline{B} \cap C) &=& (0.7)(0.5)(0.8) &=& 0.28 \\ P(\overline{A} \cap B \cap C) &=& (0.3)(0.5)(0.8) &=& 0.12 \\ P(A\cap B\cap C ) &=& (0.7_)(0.5)(0.8) &=& 0.28 \\ \hline &&&& 0.75 \end{array}\)

Therefore: .\(\displaystyle P(\text{at least 2}) \;=\;0.75\)

 

[1a2s3d4f5g6h7j8k9l]

Thank you Soroban!