A quiz consists of 10 multiple-choice questions, each with 4 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 70 %

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The minimum passing grade is `70%` means at least `7` questions must be answered correctly. At least `7` means exactly `7` or exactly `8` or exactly `9` or exactly `10.` All these possibilities are mutually incompatible, so we'll sum up their probabilities.

The probability of answering each questions correctly is `1/4,` incorrectly is `3/4.` Denote the number of `k`-combinations from `n` as `C_n^k.`

Consider answering correctly all `10` questions. The probability is `(1/4)^10` because correct or incorrect answer for the next question is independent from any previous and the probabilities of independent events are multiplied.

For exactly `9` correct answers the probability is `10*(3/4)*(1/4)^9` because one (but arbitrary) of `10` questions must be answered incorrectly (`10` possibilities) and `9` remaining correctly.

For exactly `8` it is `C_10^2*(3/4)^2*(1/4)^8,`  (there are `C_10^2` possibilities of choosing `2` questions to fail), for `7` it is  `C_10^3*(3/4)^3*(1/4)^7.`

The result may be expressed as   `sum_(k=0)^3 C_10^k*(3/4)^k*(1/4)^(10-k).`

 

In numbers it isn't so great, it is about 0.0035, or 0.35%.

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