Exploring Odds and Probability of a Symmetric Coin Toss

In summary, the conversation discusses the probability of getting an even amount of tails when tossing a symmetric coin 491 times. The speaker argues that mathematical proofs do not always require formulas and instead uses the concept of symmetry to support their answer.
  • #1
kaitoli
1
0
Hi everyone

There is a question which I find very hard to solve and it goes like this..

A symmetric coin with heads on one side and tails on the other side is tossed 491 times after one another. The total amount of times you get tails is either even or odd. Is the probability that you get an even amount of tails exactly 50%? And the question requires a strong mathematical evidence, e.g. a formula.
 
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  • #2
Mathematical proofs don't always require formulas. In this case, the basic argument is symmetry (assuming a fair coin). Therefore the probability of n tails and m heads is the same as the probability of m tails and n heads, where n and m are arbitrary with n+m=491.
 

1. What is a symmetric coin toss?

A symmetric coin toss is a random experiment in which a coin is flipped and the outcome can be either heads or tails with equal probability. This means that in a fair or ideal setting, the probability of getting heads is 50% and the probability of getting tails is also 50%.

2. How is probability calculated in a symmetric coin toss?

The probability of an event occurring in a symmetric coin toss is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In the case of a coin toss, there are two possible outcomes (heads or tails) and both have an equal chance of occurring, so the probability for each is 1/2 or 50%.

3. What is the law of large numbers in relation to a symmetric coin toss?

The law of large numbers states that as the number of trials in an experiment increases, the observed outcomes will approach their theoretical probabilities. In the case of a symmetric coin toss, as the number of tosses increases, the proportion of heads and tails will approach 50% each. This means that in the long run, the outcomes will balance out to be even.

4. Can the outcome of a symmetric coin toss be predicted?

No, the outcome of a symmetric coin toss cannot be predicted with certainty. This is because each toss is an independent event and the previous outcome does not affect the next one. In other words, the probability of getting heads or tails in each toss remains the same regardless of the previous outcomes.

5. How does a symmetric coin toss relate to real-life scenarios?

A symmetric coin toss is often used as a simple model to understand and calculate probabilities in real-life scenarios. For example, it can be used to determine the chances of winning a game of chance or the likelihood of an event occurring. Additionally, it can also be used as a starting point to understand more complex mathematical concepts such as conditional probability and Bayesian inference.

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