A password for a computer uses five digits from 0 to 9, inclusive. What is the probability that the digits of the password solely consist of prime numbers or zero? A. 1/32 B. 1/16 C. 1/8 D. 2/5 E. 1/2

(A) To calculate all the possibilities, there are 10 choices for each digit. The number of possibilities for five digits is 10 × 10 × 10 × 10 × 10 = 100,000 possible combinations.

Prime numbers constitute 2, 3, 5 and 7. In addition to zero, this is 5 total possibility for each digit. So the number of satisfactory variants is 5 × 5 × 5 × 5 × 5. So the probability is 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32. The right answer is choice (A). ------------- Please provide the formula and lesson overview which can enhance my understanding of the solution provided.

We can use general definition for probability: The number of satisfactory probabilistic events divided by the number of all possible probabilistic events.

The number of all the possible probabilistic events is 10 × 10 × 10 × 10 × 10

The number of satisfactory probabilistic events is 5 × 5 × 5 × 5 × 5

So the probability is (5 × 5 × 5 × 5 × 5) / (10 × 10 × 10 × 10 × 10) We simplify and get 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32. In this case we used only general formula, described above.

Another way is to count probability for each digit to be prime or zero: 5/10 = 1/2. And since all the 5 digits must be prime or zero at once then these probabilities are multiplied: 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32. In this case we used general formula, described above and rule for calculating probability of independent events happening at once.

Why would you want to include 0 in part of the prime numbers. I though when it is an "or" problem you would add two separate events like (probability of getting the prime number) + (probability of getting zero).

This wording implies that we consider 5 ordered digits: _ _ _ _ _ Each one of those can be either a prime number OR a zero. So the password can be: 0 2 2 7 3 0 0 0 0 0 7 2 0 5 3 5 5 7 7 2 etc.

If we had another wording, e.g. "the password consists of prime digits or it consists of zeroS", then it would imply that we consider 5 ordered digits: _ _ _ _ _ It could be 0 0 0 0 0 or 7 2 5 3 2, but it could NOT be 7 2 0 3 5.

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