1)what is the probability that a seven-digit phone number has one or more repeated digits?
2) five letters, with repetition allowed, are selected from the alphabet. what is the probability that none is repeated?
I will do the second question, they are basically both the same kind.
2) total number of ways with repetition is 26x26x26x26x26
= 26^5
number of ways with no repetition is
26x25x24x23x22
prob none repeated = 26x25x24x23x22/26^5 = .6644
in 1) you will have to subtract the number of non-repeaters from the number of repeaters first.
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To answer these probability questions, we need to first understand the total number of possible outcomes and then determine the number of favorable outcomes.
1) Probability of a seven-digit phone number having one or more repeated digits:
To determine the probability, let's first consider the total number of possible phone numbers.
- Each digit of the phone number can vary from 0 to 9 (10 options).
- Since repetition is allowed, each digit can be selected independently for each position.
Using the multiplication principle of counting, the total number of possible phone numbers is calculated as:
Total possible phone numbers = 10 * 10 * 10 * 10 * 10 * 10 * 10 = 10^7
Next, let's try to determine the favorable outcomes, i.e., phone numbers without any repeated digits.
- For the first digit, we have 10 options (0-9).
- For the second digit, we have 9 options left (excluding the already chosen digit).
- For the third digit, we have 8 options left.
- Similarly, for the remaining digits, the options decrease by 1 each time.
Using the multiplication principle again, the number of favorable outcomes is calculated as:
Number of phone numbers without any repeated digits = 10 * 9 * 8 * 7 * 6 * 5 * 4
Therefore, the probability of a seven-digit phone number having one or more repeated digits is:
Probability = 1 - (Number of phone numbers without any repeated digits / Total possible phone numbers)
2) Probability of selecting five letters from the alphabet, with repetition not allowed:
To determine the probability, let's first consider the total number of possible outcomes.
- There are 26 letters in the alphabet.
- Since repetition is not allowed, each letter can only be chosen once.
Using the permutation formula, the total number of possible outcomes is calculated as:
Total possible outcomes = P(26, 5) = 26! / (26 - 5)!
Next, let's determine the favorable outcomes, i.e., no repeated letters.
- For the first letter, we have 26 options.
- For the second letter, we have 25 options left.
- Similarly, for the remaining letters, the options decrease by 1 each time.
Using the multiplication principle, the number of favorable outcomes is calculated as:
Number of outcomes with no repeated letters = 26 * 25 * 24 * 23 * 22
Therefore, the probability of selecting five letters from the alphabet without repetition is:
Probability = Number of outcomes with no repeated letters / Total possible outcomes