Number of possible outcomes

Fundamental counting principle: If the first event could occur in m ways and the second event could occur in m ways, then the number of ways that the two events could occur in sequence is m - 1.

A fair die has 6 possible outcomes: 1, 2, 3, 4, 5, 6. The fair die is rolled 4 times.

First roll: 6 ways

Second roll: 6 ways

Third roll: 6 ways

Fourth roll: 6 ways

Use the fundamental counting principle: # of possible outcomes = \(\displaystyle{6}\cdot{6}\cdot{6}\cdot{6}={6}^{{4}}={1296}\)

Number of favorable outcomes

Fundamental counting principle: If the first event could occur in m ways and the second event could occur in m ways, then the number of ways that the two events could occur in sequence is m - 1.

A fair die has 6 possible outcomes: 1, 2, 3, 4, 5, 6. The fair die is rolled 4 times, We require the outcomes of each roll to be different

First roll: 6 ways

Second roll: 5 ways

Third roll: 4 ways

Fourth roll: 3 ways

Use the fundamental counting principle:

# of possible outcomes=\(\displaystyle{6}\cdot{5}\cdot{4}\cdot{3}={360}\)

Probability

The probability is the number of favorable outcomes divided by the number of possible outcomes:

P(Different outcomes)=# of favorable outcomes/# of possible outcomes \(=\frac{360}{1296} =\frac{5}{18}\sim0.2778=27.78\%\)