How many coin tosses does it take until you get n Heads OR m Tails?

How many coin tosses does it take until you get n Heads OR m Tails?
By Svante Janson, Lucy Martinez and Doron Zeilberger

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Written: Dec. 2025

The negative-binomial distribution famously tells you how many coin tosses would it take until you get n Heads (regardless of the number of Tails), and equivalently, how many coin tosses would it take until you get m Tails (regardless of the number of Heads). In this note we use both discrete and continuous methods to study the `hybrid' random variables: `How many coin tosses would it take until you get (for the first time) n Heads OR m Tails?, and the closely related (as we will show) random variable `How many coin tosses would it take until you get n Heads AND m Tails?'

Maple package

CoinToss.txt, a Maple package to experiment with the statistics of the number of Coin Tosses it takes until reaching n Heads OR m Tails and also the related random variable of reaching n Heads AND m Tails

Sample Input and Output for CoinToss.txt

If you want to see recurrences for the quantity L(p,h,t), for the expected number of coin tosses until you reach h Heads and t Tails where the Pr(H)=p and applying it to some special cases.
the input file yields the output file

If you want to see the linear recurrence (the same!) for the prob. generating function according to the number of tosses until you reach for the first time

n Heads OR n Tails
n Heads AND n Tails
(same recurrence, but different initial conditions) and using it in some special cases
the input file yields the output file

If you want to see two-variable linear recurrences (the same!) , in the number of desired Heads and the number of desired Tails, for the bi-variate prob. generating function according to the number of Heads and the Number of Tails reach for the first time

n Heads OR n Tails
n Heads AND n Tails
(same recurrence, but different initial conditions) and using it in some special cases
the input file yields the output file

If you want to see explicit expressions for the expectation, variance, and the first 12 central moments of the random variable "number of coin tosses of a fair coin until you get n Heads OR n Tails" AND a fully rigorous proof that the first 200 scaled central moments converge, as n goes to infinity, to those of -|N(0,1)| (the negative of the absolute value of the Normal distribution)
the input file yields the output file

Pictures
Note: the language below is in terms of family planning where Boy is Head and Girl is Tail.

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR(bottom) 100 girls look here

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR (bottom) 80 girls look here

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR (bottom) 60 girls look here

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR (bottom) 40 girls look here

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR (bottom) 30 girls look here

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR (bottom) 20 girls look here

If you want to see a plot of the expected family size as a function of the Probability of having a boy (from 0.1 to 0.9) where the goal is to have 100 boys AND (top) OR (bottom) 10 girls look here

If you want to see a plot of the probability density function for the (scaled) discrete random variable "number of coin-tosses until a fair coin reaches either 200 Heads OR 200 Tails",see here (notice how one-sided it is, and it looks like the left side of the Normal Distrubtion.
If you want to see a plot of the probability density function for the (scaled) discrete random variable "number of coin-tosses until a LOADED coin (prob. of Heads=1/3) to reach either 200 Heads OR 200 Tails", see here (notice how symmetric it is, and it looks like the full Normal Distrubtion.

Articles of Doron Zeilberger

Svante Janson's Home Page
Lucy Martinez's Home Page
Doron Zeilberger's Home Page