A thief is faced with a combination lock with five wheels. Each wheel has the
digits 0 to 9. The correct sequence of 5 digits needed to open the lock has no
repeated digits. What is the probability that the thief will guess the correct
sequence on the first try?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
10^5 = ?
since no digits are to be repeated, the number of possible valid combinations is just the number of permutations of 10 things, taken 5 at a time: 10!/5!
However, unless the burglar knows that, he still has only one chance in 10^5 of getting it right, as PsyDAG showed above.
To find the probability that the thief will guess the correct sequence on the first try, we need to determine the total number of possible sequences and the number of favorable outcomes.
Total number of possible sequences: Since there are 10 digits (0-9) on each of the five wheels, there are a total of 10^5 = 100,000 possible sequences.
Number of favorable outcomes: The thief needs to guess the correct sequence with no repeated digits. The first digit can be any of the 10 possible digits (0-9). The second digit can be any of the remaining 9 digits (since repetition is not allowed). Similarly, the third digit can be chosen from the remaining 8 digits, the fourth digit from the remaining 7 digits, and the fifth digit from the remaining 6 digits. Therefore, the number of favorable outcomes is 10 * 9 * 8 * 7 * 6 = 30,240.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible sequences:
Probability = Number of favorable outcomes / Total number of possible sequences
Probability = 30,240 / 100,000
Probability = 0.3024
Therefore, the probability that the thief will guess the correct sequence on the first try is 0.3024 or 30.24%.