# If a coin is tossed three times (or three coins are tossed together), then describe the sample space for this experiment.

Last updated date: 18th Mar 2023

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Hint: - The question is based on classic probability.Sample space of an experiment is the set of all the possible events possible in that experiment.

The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.

Given in the problem we have three coins.

If a single coin is tossed, number of outcomes will be $ = 2$

Sample Space $S = \left\{ {H,T} \right\}$

If the coin is tossed three times, no. of outcomes will be ${2^3} = 8$

Sample Space will be the cartesian product of the Set $S$ with itself twice:

$

S' = \left\{ {H,T} \right\} \times \left\{ {H,T} \right\} \times \left\{ {H,T} \right\} \\

\Rightarrow S' = \left\{ {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} \right\} \\

$

The above sample space is also valid for the experiment in which three coins are tossed at the same time.

Note: The number of outcomes in tossing $n$ number of coins $ = {2^n}$ . Always remember to check that each outcome in sample space is equally likely to occur. Also, the order of the heads or tails matter in writing sample space of coin tossing experiment.

The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.

Given in the problem we have three coins.

If a single coin is tossed, number of outcomes will be $ = 2$

Sample Space $S = \left\{ {H,T} \right\}$

If the coin is tossed three times, no. of outcomes will be ${2^3} = 8$

Sample Space will be the cartesian product of the Set $S$ with itself twice:

$

S' = \left\{ {H,T} \right\} \times \left\{ {H,T} \right\} \times \left\{ {H,T} \right\} \\

\Rightarrow S' = \left\{ {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} \right\} \\

$

The above sample space is also valid for the experiment in which three coins are tossed at the same time.

Note: The number of outcomes in tossing $n$ number of coins $ = {2^n}$ . Always remember to check that each outcome in sample space is equally likely to occur. Also, the order of the heads or tails matter in writing sample space of coin tossing experiment.

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