In the Cash 5 game of the Texas Lottery, the player picks 5 numbers from 1 to 37. The state does the same via a random draw. If the two or more match, the player wins. What is the least number of tickets you need to buy, choosing your own numbers, to ensure a win?

anonymous

The answer is 22. Below is one of many ways to guarantee a winner with 22 tickets.

• 1,2,3,4,5
• 5,6,7,8,9
• 1,2,3,8,9
• 4,6,7,8,9
• 1,2,3,6,8
• 10,11,12,13,14
• 14,15,16,17,18
• 10,11,12,17,18
• 13,15,16,17,18
• 10,11,12,15,17
• 19,20,21,22,23
• 23,24,25,26,27
• 19,20,21,26,27
• 22,24,25,26,27
• 19,20,21,24,26
• 28,29,30,31,32
• 33,34,35,36,37
• 28,30,32,34,36
• 29,31,33,35,37
• 28,30,33,35,37
• 29,31,34,36,x
• 32,33,35,37,x

Note: x = Any number

Here is my solution (PDF).

This question is asked and discussed in my forum at Wizard of Vegas.

What is the probability of losing 25 out of 25 pass line bets in craps? You may ignore bets that resolve on the come out roll.

Jeff

Assuming a point is made, here are the ways it could be made:

• Point of 4: 1-3, 3-1, 2-2 = 3 ways
• Point of 5: 1-4, 4-1, 2-3, 3-2 = 4 ways
• Point of 6: 1-5, 5-1, 2-4, 4-2, 3-3 = 5 ways
• Point of 8: 2-6, 6-2, 3-5, 5-3, 4-4 = 5 ways
• Point of 9: 3-6, 6-3, 4-5, 5-4 = 4 ways
• Point of 10: 4-6, 6-4, 5-5 = 3 ways

The total number of ways is 24.

The probability of not making a point of 4 or 10 is 6/9 = 2/3. The probability of not making a point of 5 or 9 is 6/10 = 3/5. The probability of not making a point of 6 or 8 is 6/11.

The overall probability of losing, assuming a point was establish is thus:

Prob(Point is 4) × Prob(Roll 7 before 4) + Prob(Point is 5) × Prob(Roll 7 before 5) + Prob(Point is 6) × Prob(Roll 7 before 6) + Prob(Point is 8) × Prob(Roll 7 before 8) + Prob(Point is 9) × Prob(Roll 7 before 9) + Prob(Point is 10) × Prob(Roll 7 before 10)

= (3/24) × (2/3) + (4/24) × (3/5) + (5/24) × (6/11) + (5/24) × (6/11) + (4/24) × (3/5) + (3/24) × (2/3) = 98/165 =~ 0.593939394.

The probability of losing this way 25 times in a row is (98/165)25 =~ 0.000002205735 =~ 1 in 453,364.

What is the closest planet to the earth, on average?

anonymous

I note the "on average" in your question. If you had asked which planet gets the closest to earth when it passes, known as the "approach," the answer would be Venus. However, "on average" includes all points in the orbits of both planets, including when the two planets are on opposites sides of the sun.

To answer your question, I'm going to assume all planets are in the same plane and travel in circular orbits. While neither is exactly the case, the answer is overwhelming in favor of Mercury that any errors due to these assumptions being wrong are negligible in comparison.

The following table shows each planet's distance from the sun as well as the closest, furthest, and average distance from earth. All distances are in Astronomical Units, which is the distance of the Earth from the Sun. As you can see, Mercury has the least average distance at 1.038401 AU's.

Closest Planet

Planet Distance
from Sun
Cloest
Distance
from Earth
Furthest
Distance
from Earth
Average
Distance
from Earth
Mercury 0.39 0.61 1.39 1.038401
Venus 0.72 0.28 1.72 1.134494
Mars 1.52 0.52 2.52 1.689509
Jupiter 5.2 4.2 6.2 5.248189
Saturn 9.54 8.54 10.54 9.566223
Uranus 19.2 18.2 20.2 19.213023
Neptune 30.06 29.06 31.06 30.068317

Source for distances from the sun: NASA.

The above averages were taken by averaging the distance over the 360 different degrees the other planet could be around the sun and assuming a fixed location for the earth.

For further reading, I recommend Venus is not Earth’s closest neighbor from Physics Today.

In Ultimate Texas Hold 'Em, if the player makes the Trips bet and later folds a three of a kind or better, does the Trips bet still win?

anonymous

Yes!

The Trips bet is supposed to play, even if the dealer folds. I know some dealers and floormen incorrectly think a Play bet is required for the Trips to have action, but they are wrong.

Sources: