Calculate the probabilities that the first digit of the zip code is 3, the second digit is 9, and so on.
≈ 0.002
Practice makes perfect
We are asked to find the probability that a zip code made randomly from among the digits 3, 7, 3, 9, 5, 7, 2, and 3 is the number 39372. To do that, we will first calculate the probabilities that the first digit of the zip code is 3, the second is 9, and so on.
First Digit
Let's find the probability that the first digit of the zip code is 3. Let's look at the possible digits.
3,7, 3,9,5,7,2, 3
We can see that out of 8 possible digits, only 3 of them are the digit we want. Therefore the probability that the first digit is 3 is 3 8.
P(First digit is3)=3/8
Second Digit
Now, we can find the probability that the second digit of the zip code is 9. This event and choosing the first digit are dependent events. Therefore, we will have to take into account that the first digit of the zip code is 3. Let's look at the remaining digits.
3,7,3, 9,5,7,2,3
Because the first digit of the zip code is already set, we have 7 possible digits left. Only 1 of the digits is equal to 9. Therefore, the probability that the second digit is 9 is equal to 1 7.
P(Second digit is9)=1/7
Rest of the Digits
We will repeat this process to find the probabilities of the remaining zip code digits. Let's summarize the results in a table.
Digit of the Zip Code
Possible Digits
Probability
First, 3
3,7, 3,9,5,7,2, 3
3/8
Second, 9
3,7,3, 9,5,7,2,3
1/7
Third, 3
3,7, 3,9,5,7,2, 3
2/6
Fourth, 7
3, 7,3,9,5, 7,2,3
2/5
Fifth, 2
3,7,3,9,5,7, 2,3
1/4
Finally, to find the probability that the zip code is 39372, we need to multiply all the probabilities from the last column.
P(Zip code is39372)=3/8*1/7*2/6*2/5*1/4 ≈ 0.002
The probability is equal to approximately 0.002.
Alternative Solution
Permutations
We will find the probability that a randomly generated zip code from the given digits is the number 39372. To do that, we will compare the number of favorable outcomes with possible outcomes.
P=Favorable outcomes/Possible outcomes
First, we can calculate the number of possible outcomes, which is the number of possible zip codes that we can generate using the given digits.
Given digits:3,7,3,9,5,7,2,3
We will sort them for clarity.
Given digits:2,3,3,3,5,7,7,9
There are repeats among the digits and this makes calculating the number of possible zip codes tricky. Therefore, to make calculations simpler we can differentiate the repeated digits by, for example, coloring the repeated digits using different colors. 2, 3, 3, 3,5, 7, 7,9
Each of the 8 digits is now different. Let's calculate the number of possible zip codes. Since the order of the digits matters, we will use permutations.
Permutations
The number of permutations of n distinct objects taken r at a time is denoted by _()brownnP_()#1DB78Br and can be calculated using the following formula.
_()brownnP_()#1DB78Br=n!/(n-r)!
We have a total of 8 digits and we need to select 5 of them to create a zip code. Therefore, to calculate the number of zip codes, we will substitute n=8 and r=5.
We found that the number of possible zip codes is 6720. The favorable outcome is when the generated zip code is the number 39372.
We have three options for the digit 3.
3, 3,or 3
There are two 3's in the zip code, so we can select them and their order in _3P_2 different ways. Similarly, the digit 7 can be chosen in _2P_1 different ways.
7or 7
The rest of the digits have no repeats. Therefore, the number of favorable outcomes is _3P_2* _2P_1. Let's use the Permutation Formula again to calculate the number of favorable outcomes.
Favorable outcomes= _3P_2* _2P_1
⇕
Favorable outcomes=3!/1!* 2!/1!
Let's simplify this expression!
The number of favorable outcomes is 12. Earlier we found that the number of possible outcomes is 6720. Finally, we are ready to calculate the probability that the randomly generated zip code is the number 39372.
P=Favorable outcomes/Possible outcomes=12/6720 ≈ 0.002
