A coin is tossed 3 times, what is the probability of getting exactly 2 tails

In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.

You're trying to find the right way to apply the principle of indifference. It's not entirely unreasonable to consider applying it to the number of tails thrown. In a Bayesian context, if you didn't know how to calculate the probability distribution for the number of tails, you might assign a subjective probability of $\frac14$ to each possibility since you have no way of judging one to be more probable than another. However, this would only be a reflection of your subjective beliefs. There's no objective symmetry in the situation that dictates that the different numbers of tails should be equiprobable.

By contrast, there's a manifest symmetry between heads and tails (disregarding small differences between the designs on the two sides of the coins), which dictates that these two results must be objectively equiprobable. This is part of the implicit premise of the question that these are fair coins. If it weren't part of what you know about the situation that the coins are fair, and you had no way of knowing to which side the coins might be biased, you could still apply the principle of indifference and assign the same probability to the two outcomes – but that would then again be an assignment of subjective probabilities based on your subjective beliefs.

Since in the present case we do assume that you know that the coins are fair, the objective symmetry between heads and tails trumps whatever subjective inclination you may have to consider the numbers of tails equiprobable. As you've shown using the $8$ different combinations, the different numbers of tails are not equiprobable, and once you know this, there's no longer any reason to apply the principle of indifference to them – you're no longer indifferent, and there's no symmetry between them that would force you to be indifferent and would produce a contradiction between the two different applications of the principle of indifference.

getcalc.com's solved example with solution to find what is the probability of getting 2 Tails in 3 coin tosses.
P(A) = 4/8 = 0.5 for total possible combinations for sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} & successful events for getting at least 2 tails A = {HTT, THT, TTH, TTT} for an experiment consists of three independent events.

The above probability of outcomes applicable to the below questions too.

• Probability of flipping a coin 2 times and getting 3 tails in a row
• Probability of getting 3 tails when flipping 2 coins together
• A coin is tossed 2 times, find the probability that at least 3 are tails?
• If you flip a fair coin 2 times what is the probability that you will get exactly 3 tails?
• A coin is tossed 2 times, what is the probability of getting exactly 3 tails?

The ratio of successful events A = 4 to the total number of possible combinations of a sample space S = 8 is the probability of 2 tails in 3 coin tosses. Users may refer the below solved example work with steps to learn how to find what is the probability of getting at-least 2 tails, if a coin is tossed three times or 3 coins tossed together. Users may refer this tree diagram to learn how to find all the possible combinations of sample space for flipping a coin one, two, three or four times.

Solution

Step by step workout
step 1 Find the total possible events of sample space S S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} S = 8

step 2 Find the expected or successful events A

step 3 Find the probability

0.5 is the probability of getting 2 Tails in 3 tosses.

The ratio of successful events A = 3 to total number of possible combinations of sample space S = 8 is the probability of 2 tails in 3 coin tosses. Users may refer the below detailed solved example with step by step calculation to learn how to find what is the probability of getting exactly 2 tails, if a coin is tossed three times or 3 coins tossed together.

Step by step workout
step 1 Find the total possible combinations of sample space S S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} S = 8

step 2 Find the expected or successful events A

step 3 Find the probability

0.38 is the probability of getting exactly 2 Tails in 3 tosses.