A student claims that if a fair coin is tossed and comes
up heads 5 times in a row, then, according to the law
of averages, the probability of tails on the next toss
is greater than the probability of heads. What is your
reply?
No matter what comes up or how often you get heads, the chance for each individual throw is always the same -- 1 chance out of 2.
The previous tosses do not effect later tosses; they are independent.
http://www.members.cox.net/dagershaw/lol/Odds.html
My reply is that the student's claim is not correct. According to the law of averages, also known as the law of large numbers, as the number of trials increases, the observed results should converge towards the theoretical probability. In this case, the theoretical probability of getting heads or tails on a fair coin toss is always 0.5 or 50%.
Each coin toss is an independent event, and the outcome of each toss does not depend on the previous tosses. So, even if the coin has landed heads 5 times in a row, the probability of getting heads on the next toss is still 0.5. The previous outcomes do not affect the future outcomes.
To reiterate, the law of averages does not imply that if one outcome has occurred multiple times, the other outcome is more likely to occur next. The probability remains the same for each coin toss, regardless of the previous results.