There are 100 Snakes on the Plane. There are 100 Passengers on the Plane. Each Snake will Bite One Passenger at Random. What is the Probability that The Target, Mr. X, does not get bit?
A Sadly Limited Way of Looking at This
1. Suppose the Snakes are Labeled as 1, 2, 3
2. Suppose the People are Labeled as X, Y, Z
3. Looking at a Sample Space, here is what could happen. We put numbers for the Snake(s) that Bite the Person
| X | Y | Z |
|---|---|---|
| 1, 2, 3 | - | - |
| 1, 2 | 3 | - |
| 1, 2 | - | 3 |
| 2, 3 | 1 | - |
| 2, 3 | - | 1 |
| 1, 3 | 2 | - |
| 1, 3 | - | 2 |
| 1 | 2, 3 | - |
| 1 | - | 2, 3 |
| 1 | 2 | 3 |
| 1 | 3 | 2 |
| 2 | 1, 3 | - |
| 2 | - | 1, 3 |
| 2 | 1 | 3 |
| 2 | 3 | 1 |
| 3 | 1, 2 | - |
| 3 | - | 1, 2 |
| 3 | - | 1, 2 |
| 3 | 1 | 2 |
| 3 | 2 | 1 |
| - | 1, 2, 3 | - |
| - | - | 1, 2, 3 |
| - | 1, 2 | 3 |
| - | 3 | 1, 2 |
| - | 1, 3 | 2 |
| - | 2 | 1, 3 |
| - | 2, 3 | 1 |
| - | 1 | 2, 3 |
4. Notice that there are 27 total Possibilities. And, 8 of the 27 possibilities do not have any Bites for Person X. So, P(Person X not bit) = 8/27 ≈ 29.6 %
We weep because that style will be cumbersome for 4 Snakes & People, or higher. We could look at the Case of 3 Snakes and People in the following way:
- Suppose the Snakes are Labeled as S1, S2, S3
- Let’s look for a moment at Snake S1:
- P(Snake S1 Bites Mr. X) = 1/3
- In other words, there’s a 1/3 chance that Mr. X will get bitten by that particular Snake, S1
- P(Snake S1 does not Bite Mr. X) = 1 – 1/3
- This is the complement of the bullet above
- P(Snake S1 Bites Mr. X) = 1/3
- Notice that the logic of [2] applies to each Snake. That is,
- P(Snake S1 does not Bite Mr. X) = 1 – 1/3
- P(Snake S2 does not Bite Mr. X) = 1 – 1/3
- P(Snake S3 does not Bite Mr. X) = 1 – 1/3
- Thus, P(Mr. X does not get Bitten ) = The Product of all the Probabilities in [3],
- (1 – 1/3) x (1 – 1/3) x (1 – 1/3)
- S1 no Bite & S2 no Bite & S3 no Bite
- So, the Final Solution for P(Mr. X Not Bit) = (1 – 1/3)3 = 8/27 ≈ 29.6 %
*Note* that we are assuming independence in Step [4], we’ll leave it to the Biologists to interpret whether this is a reasonable assumption
