Final answer:
The probability of a 4-digit PIN beginning with a digit greater than 6 and ending with a 5, with no restrictions on the middle digits, is 1/3 or approximately 33.33%.
Step-by-step explanation:
To find the probability of a 4-digit PIN beginning with a digit greater than 6 and ending with 5, we must consider each position of the PIN independently, the first digit can be either 7, 8, or 9 by the given restriction, and there are 3 possibilities out of 9 acceptable starting digits (1-9). For the remaining two middle digits, there are 10 possibilities for each (0-9), since there are no restrictions, and the last digit must be a 5, so there is only 1 possibility out of 10.
Hence, you calculate the probability as follows:
- First digit: 3 possibilities (7, 8, or 9)
- Middle two digits: 10 possibilities each
- Last digit: 1 possibility (must be 5)
The total number of 4-digit PINs starting with any digit except zero is 9*10*10*10 (since the first digit can be any from 1 to 9 and each of the remaining three digits can be any from 0 to 9). The number of favourable outcomes is 3*10*10*1. The probability is the number of favourable outcomes divided by the total number of possible outcomes:
Probability = (3*10*10*1) / (9*10*10*10) = 3/9 = 1/3.
Therefore, the probability of a 4-digit PIN starting with a digit greater than 6 and ending with a 5 is 1/3 or approximately 33.33%.