# Coin Flip winning probability increase in sequence

Flipping a coin is an independent event, and has a chance of 50%. So Lets say that I flip a coin 10 times, and if the first 5 flips are heads. Then the probability of the next flip being a tails, is higher(so as to satisfy the 50% chance), How can we calculate the increase in probability of this happening.

I read an article about a roulette wheel. In a roulette wheel the reds and blacks have an equal chance of hitting.(ignoring the zero) just like a coin flip. Now the article said that if you place a bet on red, then you have a 52% chance of losing(as this includes the zero).

If you bet on red again, then you have only a 28% chance of losing in the second consecutive bet.

And the chances of a red coming keeps increasing each flip.

My Question is that, since these events are independent, then shouldnt the probability of me winning or losing be 50% everytime. May it be 10 consecutive blacks..

Also, if to satisfy the 50% over a large sample size, the probability of red hitting does increase, then how do we calculate it?

• You started by saying "Flipping a coin is an independent event" but then said "Then the probability of the next flip being a tails, is higher". Which is it? Either the tosses are independent, or one toss depends on the previous ones. It's hard to answer the question when the third sentence contradicts the first. – Dan Piponi Jun 11 '14 at 0:52
• possible duplicate of Past coin tosses affect the latest one if you know about them? – MJD Jun 11 '14 at 2:29

The article you reference is perhaps a little misleading. The probability of losing if you bet on red stays 52% each individual time you roll, but the chance that you lose $\textit{every time}$ naturally decreases the more times you roll. In fact if you lose on a roll with probability $p$, then the probability that you have lost every time by the $n$-th roll is $p^n$. Note that this gives probabilities of losing as $.52^2\approx.27$ for the second roll and $.52^3\approx.14$ for the third roll and so on (I think probably the probability of losing should actually be $\frac{20}{38}\approx.526$ which gives numbers consistent with the table you gave).
The key point here is that if you have truly independent events, then the outcomes of previous events $\textit{do not affect}$ the likelihoods of future outcomes. In the case of coin flipping, the tendency of large samples to have approximately half heads and half tails just comes from having many repeated trials. The mistake that previous outcomes affect future outcomes is commonly called the gambler's fallacy.