Question 14.7: Selecting a Key A person selects a key at random from four k......
Selecting a Key
A person selects a key at random from four keys to open a lock. Only one key opens the lock. If the first key does not open the lock, she tries other keys until one opens the lock. Find the average of the number of keys a person will have to try to open the lock. Try the experiment 25 times.
Step 1 List all possible outcomes of the experiment. They are key 1, key 2 , key 3 , and key 4.
Step 2 Determine the probability of each outcome. Since a key is selected at random and there are four keys, the probability of selecting each key is \frac{1}{4}.
Step 3 Set up a correspondence between the random numbers and the outcomes. Assume that each key is numbered from 1 through 4 and that key 2 opens the lock. Naturally, the person doesn’t know this, so she selects the keys at random. For the simulation, select a sequence of random digits, using only 1 through 4 , until the digit 2 is reached. The trials are shown here.
Step 4 Select random numbers from the table and repeat the experiment 24 times.
Step 5 Compute any statistics and state the conclusions.
Find the average:
\bar{X}=\frac{\sum X}{n}=\frac{1+1+\cdots+3}{25}=\frac{54}{25}=2.16
The theoretical average is 2.5. Again, only 25 repetitions were used; more repetitions should give a result closer to the theoretical average.
| Trial | Random digit (key) | Number | Trial | Random digit (key) | Number |
| 1 | 2 | 1 | 14 | 2 | 1 |
| 2 | 2 | 1 | 15 | 4 2 | 2 |
| 3 | 1 2 | 2 | 16 | 1 3 2 | 3 |
| 4 | 1 4 3 2 | 4 | 17 | 1 2 | 2 |
| 5 | 3 2 | 2 | 18 | 2 | 1 |
| 6 | 3 1 4 2 | 4 | 19 | 3 4 2 | 3 |
| 7 | 4 2 | 2 | 20 | 2 | 1 |
| 8 | 4 3 2 | 3 | 21 | 2 | 1 |
| 9 | 4 2 | 2 | 22 | 2 | 1 |
| 10 | 2 | 1 | 23 | 4 2 | 2 |
| 11 | 4 2 | 2 | 24 | 4 3 1 2 | 4 |
| 12 | 3 1 2 | 3 | 25 | 3 1 2 | 3 |
| 13 | 3 1 2 | 3 | Total 54 |