Draw probability - last ballot wins

Draws traditionally work where the first ballot pulled is the winner. In this, the chances of winning are easily predicted.

However, if a draw is drawn out for more suspense, and the LAST ballot remaining is the winner do your chances increase or decrease compared to the traditional way of drawing (first ballot wins)? In this case an individual can have multiple ballots in the draw.


For example, if I had 10 ballots with my name on them in a box of 40 ballots total, my chances of winning on the first ballot are 25%, but what are my chances of being the last ballot pulled? Is it higher than 25% or lower?

Thanks!

Hello, adammiron!

Draws traditionally work where the first ballot pulled is the winner.
In this, the chances of winning are easily predicted.

However, if a draw is drawn out for more suspense, and the LAST ballot remaining is the winner,
do your chances increase or decrease compared to the traditional way of drawing (first ballot wins)?
In this case an individual can have multiple ballots in the draw.

For example, if I had 10 ballots with my name on them in a box of 40 ballots total,
my chances of winning on the first ballot are 25%.,
But what are my chances of being the last ballot pulled? .Is it higher than 25% or lower?

Click to expand...

Note that the first 39 draws are ignored.

The probability that the 40th draw is one of your ballots is: \(\displaystyle \frac{10}{40} \,=\,25\%\)


Consider a similar problem.

A standard deck of cards is shuffled.

What is the probability that the top card is the \(\displaystyle J\spadesuit\,?\)
. . \(\displaystyle P(J\spadesuit,\,1^{st}) \:=\:\frac{1}{52}\)

What is the probability that the bottom card is the \(\displaystyle J\spadesuit\,?\)
. . \(\displaystyle P(J\spadesuit,\,52^{nd}) \:=\:\frac{1}{52}\)

What is the probability that the 19th card is the \(\displaystyle J\spadesuit\,?\)
. . \(\displaystyle P(J\spadesuit,\,19^{th}) \:=\:\frac{1}{52}\)


Got it?

adammiron said:

Draws traditionally work where the first ballot pulled is the winner. In this, the chances of winning are easily predicted.

However, if a draw is drawn out for more suspense, and the LAST ballot remaining is the winner do your chances increase or decrease compared to the traditional way of drawing (first ballot wins)? In this case an individual can have multiple ballots in the draw.


For example, if I had 10 ballots with my name on them in a box of 40 ballots total, my chances of winning on the first ballot are 25%, but what are my chances of being the last ballot pulled? Is it higher than 25% or lower?

Thanks!

Click to expand...

The probability of your name being chosen first is the number of possible orderings of 40 ballots, in which your name appears first, divided by the total number of possible orderings. Similary, the probability of your name being chosen last is the number of possible orderings of 40 ballots, in which your name appears last, divided by the total number of possible orderings

Given any possible ordering for the draws of the 40 ballots, the reversal is also a possible ordering. Further, to every possible ordering with you name chosen first, there corresponds the reverse ordering in which your name is the last chosen. That is the two fractions have exactly the same numerator and the same denominator. The probabilities either way are the same.

(And, of course, if you were to say that the "winner" will be the nth name chosen, for fixed n between 1 and 40, you would still get exactly the same probability.)