Ask The Wizard #337
You are given:
- Imagine there's an aircraft flying eight miles above a surface-to-air missile, which is currently in a targeted position. fired at that moment.
- Throughout its flight, the aircraft consistently moves in a straight line.
- The airplane travels at 600 miles per hour.
- The missile travels at 2000 miles per hour.
- The missile, on the other hand, is always oriented at a trajectory that aims directly at the aircraft.
Questions:
- What distance will the aircraft cover before the missile makes contact?
- What is the duration required for the missile to reach the aircraft?
- What distance will the missile travel during its course?
- Distance covered by the aircraft before being hit by the missile is 240/91 miles.
- The time it takes for the missile to impact the aircraft is 2/455 hours.
- The missile's travel distance amounts to 800/91 miles.
Here is my solution (PDF).
This topic is currently under discussion on my forum at Wizard of Vegas .
I once witnessed an astonishing sequence of 49 consecutive hands in baccarat, where 48 ended in Player victories, excluding ties. What are the odds of that happening in a single shoe?
Typically, a shoe consists of approximately 80.884 total hands. Given that the probability of a Tie is 0.095156, we can remove those from consideration and anticipate around 73.18740 hands per shoe, ignoring ties.
The odds of any set of 49 consecutive hands, again ignoring ties, resulting in 48 wins for the Player is a staggering 1 in 21,922,409,835,345. Moreover, there are about 25.1874 potential starting points for these 49 hands, allowing us to make an estimation. Therefore, the likelihood of witnessing this event within a shoe is approximately 1 in 870,371,922,467. While this figure may not be exact, I believe it serves as a solid estimate.
Assume:
- 90% of the public wears masks.
- The chance of contracting the coronavirus stands at 1% for individuals wearing masks and 3% for those who do not.
If someone is randomly selected and has the coronavirus, what are the odds that they wear a mask?
This scenario illustrates a traditional Bayesian conditional probability problem.
The solution can be expressed as probability(somebody is a mask-wearer and has coronavirus)/probability(somebody has coronavirus) =
(0.9*0.01) / (0.9*0.01 + 0.1*0.03) = 75%.
When rolling a fair die ten times, what are the chances that the outcome forms a non-decreasing sequence? In other words, each result must be equal to or exceed the previous roll.
The solution is complex but can be determined by recursively calculating the probability that the last roll of any sequence is x, ensuring each result meets or surpasses the previous one.
The table below illustrates the probabilities for roll outcomes ranging from 1 to 10, with the final side rolled varying from 1 to 6.
Ten Non-Decreasing Rolls
Roll Number | Side of 1 | Side of 2 | Side of 3 | Side of 4 | Side of 5 | Side of 6 | Total |
---|---|---|---|---|---|---|---|
1 | 0.166667 | 0.166667 | 0.166667 | 0.166667 | 0.166667 | 0.166667 | 1.000000 |
2 | 0.027778 | 0.055556 | 0.083333 | 0.111111 | 0.138889 | 0.166667 | 0.583333 |
3 | 0.004630 | 0.013889 | 0.027778 | 0.046296 | 0.069444 | 0.097222 | 0.259259 |
4 | 0.000772 | 0.003086 | 0.007716 | 0.015432 | 0.027006 | 0.043210 | 0.097222 |
5 | 0.000129 | 0.000643 | 0.001929 | 0.004501 | 0.009002 | 0.016204 | 0.032407 |
6 | 0.000021 | 0.000129 | 0.000450 | 0.001200 | 0.002701 | 0.005401 | 0.009902 |
7 | 0.000004 | 0.000025 | 0.000100 | 0.000300 | 0.000750 | 0.001650 | 0.002829 |
8 | 0.000001 | 0.000005 | 0.000021 | 0.000071 | 0.000196 | 0.000472 | 0.000766 |
9 | 0.000000 | 0.000001 | 0.000004 | 0.000016 | 0.000049 | 0.000128 | 0.000199 |
10 | 0.000000 | 0.000000 | 0.000001 | 0.000004 | 0.000012 | 0.000033 | 0.000050 |
The answer can be found in the bottom right corner, and if we expand it to more decimal places, the result is approximately 0.0000496641295788244, which translates to about 1 in 20,135.
This topic is currently under discussion on my forum at Wizard of Vegas .