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Ask the Wizard #327

If you flip a fair coin 40 times, what is the probability of getting BOTH a streak of 5 heads and a streak of 5 tails? “A streak” meaning at least 1 and “5” meaning at least 5. The head and tail streaks do not have to be adjacent.

Ace2

Please click the button below for the answer.

The answer is 107,094,548,225 / 549,755,813,888 = apx. 19.4804%.

Here is my solution (PDF).

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

What is the standard deviation in craps, assuming a pass line bet and taking 3-4-5x odds? How about a don't pass bet and laying 3-4-5x odds?

"Anonymous" .

The standard deviation, relative to the pass bet, with full 3-4-5x odds is 4.915632.

The standard deviation, relative to the don't pass bet, laying full 3-4-5x odds is 4.912807.

A microbe, let’s call it Covid-20 can spawn a new microbe at any time. The probability of a particular microbe spawning at any given time, from a specific parent microbe, is always the same, regardless of the time since the last spawning. The average time between spawnings from the same microbe is one day. In mathematical terms, the expected time between spawnings from the same microbe follows an exponential distribution with a mean of one day.

Once a microbe enters your lungs, what is the expected number of microbes will you have after seven days?

"Anonymous" .

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The answer is e7 = apx. 1,096.6332.

Click the button below for the solution.

This solution will require an ordinary differential equation. If you're not at that point yet in your math education, you won't get it.

Let:
m = Number of Covid-20 microbes
t = time, in days

Since each microbe average a new microbe once per day, m microbes will average m new microbes per day. In other words, the rate of increase in microbes (m) at any given time t can be written as:

dm/dt = m.

I'm not sure the proper way to express this, but separate the dt to the right side:

dm = m dt.

Divide both sides by m:

1/m dm = 1 dt.

Integrate both sides:

ln(m) = t + C, where C is the constant of integration.

We're given that at time 0 there is one microbe. In other words when t = 0, m = 1. We can put those values in the equation above to solve for C:

ln(1) = 0 + C

0 = 0 + C

C = 0.

We now have ln(m) = t.

Take the exp() of both sides:

m = et

So, at time t=7, there will be e7 = apx. 1096.6332 microbes.

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

What is the player advantage in Ultimate Texas Hold 'Em if the player is not required to make a Blind bet?

Eliot from Santa Barbara

This is a good question because some dealers have been known to not enforce the Blind bet rule. The Blind bet has a huge house advantage, so not having to make it would be very beneficial to the player.

Assuming the player follows optimal strategy based on the correct rules (required a Blind bet), then the player advantage would be 29.28%. It would be even higher following a strategy based on no Blind bet being required.