Paula moves to an area with a different telephone exchange. Telephone numbers in the new exchange start with 753, and all combinations of the four remaining digits are equally likely.
a) Calculate the probability that the last four digits in Paula’s new telephone number are odd digits.
b) What is the expected number of odd digits in her new telephone number?
Number of ways to pick 4 random even digits = 5^4 = 625
Number of ways to pick 4 random digits = 10^4 10000
So prob. of all even = 625/10000 = 1/16
The expected (i.e. most likely) number of even digits is 2. This is because the peak of the binomial distribution occurs for 2 even digits. You have a 4/16 = 1/4 probability of 1 even digit, a 4/16 = 1/4 probability of 3 even digits, a 1/16 probability of no even digits and thus a 6/16 = 3/8 probability of 2 even digits. These numbers come straight from Pascal's triangle 1 4 6 4 1
a) To calculate the probability that the last four digits in Paula's new telephone number are odd digits, we need to determine the total number of possible combinations of the last four digits and the number of favorable combinations where all four digits are odd.
There are 10 possible digits (0-9), and since we know that all combinations are equally likely, each digit has a 1/10 chance of being chosen.
Since all four digits in the telephone number need to be odd, we need to calculate the probability for each digit to be odd, then multiply them together.
The probability of a digit being odd is 5/10 (since half of the digits, 0, 2, 4, 6, and 8, are even).
Therefore, the probability of the last four digits in Paula's new telephone number being odd digits is (5/10) * (5/10) * (5/10) * (5/10) = 625/10,000 = 0.0625.
So the probability is 0.0625 or 6.25%.
b) To calculate the expected number of odd digits in Paula's new telephone number, we need to calculate the probability of having different numbers of odd digits (0 to 4) and then multiply those probabilities by the corresponding number of odd digits.
The probability of having 0 odd digits is (5/10) * (5/10) * (5/10) * (5/10) = 625/10,000 = 0.0625.
The expected number of odd digits in this case is 0.
The probability of having 1 odd digit is (5/10) * (5/10) * (5/10) * (5/10 - 1/10) * 4 = 500/10,000 = 0.05.
The expected number of odd digits in this case is 1.
Similarly, calculating the probabilities for having 2, 3, and 4 odd digits:
The probability of having 2 odd digits is (5/10) * (5/10) * (5/10 - 1/10) * (4/10) * 4 = 375/10,000 = 0.0375.
The expected number of odd digits in this case is 2.
The probability of having 3 odd digits is (5/10) * (5/10 - 1/10) * (4/10) * (1/10) * 4 = 100/10,000 = 0.01.
The expected number of odd digits in this case is 3.
The probability of having 4 odd digits is (5/10 - 1/10) * (4/10) * (1/10) * (1/10) * 4 = 20/10,000 = 0.002.
The expected number of odd digits in this case is 4.
Now, we multiply each expected number by its corresponding probability and sum them up:
(0 * 0.0625) + (1 * 0.05) + (2 * 0.0375) + (3 * 0.01) + (4 * 0.002) = 0 + 0.05 + 0.075 + 0.03 + 0.008 = 0.163.
Therefore, the expected number of odd digits in Paula's new telephone number is 0.163 or approximately 0.163.
a) To calculate the probability that the last four digits in Paula's new telephone number are odd, we need to determine the total number of possible combinations and the number of combinations that satisfy the given condition.
1. Total number of possible combinations:
Since the new telephone numbers start with 753 and all remaining four digits can be any number, we have 10 options for each of the four digits (0-9). Therefore, the total number of possible combinations is 10 * 10 * 10 * 10 = 10,000.
2. Number of combinations with all four digits odd:
For each digit ending in an odd number, there are 5 odd digits (1, 3, 5, 7, 9) out of 10 possible digits. Since there are four remaining digits, the number of combinations with all four digits odd is 5 * 5 * 5 * 5 = 625.
3. Probability calculation:
To calculate the probability, we divide the number of combinations with all four digits odd by the total number of possible combinations:
P = Number of combinations with all four digits odd / Total number of possible combinations
P = 625 / 10,000
P = 0.0625
Therefore, the probability that the last four digits in Paula's new telephone number are odd is 0.0625 or 6.25%.
b) To find the expected number of odd digits in Paula's new telephone number, we need to determine the probability of having 0, 1, 2, 3, or 4 odd digits and then calculate the sum of the products of each possible outcome and its corresponding probability.
1. Probability of having 0 odd digits:
In this case, all four digits in the telephone number would have to be even. There are 5 even digits (0, 2, 4, 6, 8), so the probability is (5/10) * (5/10) * (5/10) * (5/10) = 0.0125.
2. Probability of having 1 odd digit:
There are 4 ways to choose which digit will be odd (4C1), and 5 ways to choose the odd digit itself (5 options). The remaining three digits would be even, so the probability is (4/10) * (5/10) * (5/10) * (5/10) = 0.125.
3. Probability of having 2 odd digits:
In this case, there are 6 ways to choose which two digits will be odd (4C2), and 5 options for each odd digit. The remaining two digits would be even, so the probability is (6/10) * (5/10) * (5/10) * (5/10) = 0.1875.
4. Probability of having 3 odd digits:
Similar to the previous cases, there are 4 ways to choose which three digits will be odd (4C3), and 5 options for each odd digit. The remaining one digit would be even, so the probability is (4/10) * (5/10) * (5/10) * (5/10) = 0.125.
5. Probability of having 4 odd digits:
In this case, there is only one way for all four digits to be odd, which is (5/10) * (5/10) * (5/10) * (5/10) = 0.0625.
Now, we calculate the expected value by summing up the products of each outcome and its corresponding probability:
Expected value = (0 * 0.0125) + (1 * 0.125) + (2 * 0.1875) + (3 * 0.125) + (4 * 0.0625)
Expected value = 0 + 0.125 + 0.375 + 0.375 + 0.25
Expected value = 1.125
Therefore, the expected number of odd digits in Paula's new telephone number is 1.125.