HW 14 Equally Likely outcomes Conditional Probability and In

HW 1.4. (Equally Likely outcomes, Conditional Probability, and Independence A computer system uses passwords that contain exactly six characters, and each character is one of the 26 lowercase letters (a-z) or 26 uppercase letters (A-Z) or 10 integers (C-9). Let S denote the set of all possible password, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in S are equally likely 1. How many outcomes in sample space S? 2. How many outcomes in event A? 3. How many outcomes in event B? 4. How many outcomes in event An B? 5. POA) 6. P(AIB) 7. Are events A and B independent? Why? Bonus! P(password contains exactly 2 integers given that it contains at least 1 integer

Solution

1.

For each character, there are 26+26 + 10 = 62 possibilities.

As they can repeat, then there are 62^6 = 56800235584 OUTCOMES in sample S. [answer]

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2.

For each character, there are 26+26 = 52 possibilities.

As they can repeat, then there are 52^6 = 19770609664
OUTCOMES in event A. [answer]

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3.

For each character, there are 10 possibilities.

As they can repeat, then there are 10^6 = 1000000
OUTCOMES in event B. [answer]

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4.

ZERO. Events A and B do not intersect as event A is all letters, while B is all numbers.

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5.

P(A) = 19770609664/56800235584 = 0.34807267 [answer]

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6.

P(A|B) = P(A and B) / P(B)

as P(A and B) = 0 [mutually exclusive]

P(A|B) = 0 [answer]

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7.

No, as P(A) =/ P(A|B).

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