How many coin tosses does it take until you get r Heads OR s Tails?
By Svante Janson, Lucy Martinez and Doron Zeilberger
.tex [YET TO BE WRITTEN]
.pdf
Written: Nov. 2025
Maple package
-
CoinToss.txt,
a Maple package to experiment with the statistics of the number of Coin Tosses it takes
until reaching r Heads or s 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" then
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
Articles of Doron Zeilberger
Svante Janson's Home Page
Lucy Martinez's Home Page
Doron Zeilberger's Home Page
