Final answer:
The theoretical probability of rolling a 1 or 6 on a fair die and flipping tails on a fair coin is 1/6. This is determined by multiplying the separate probabilities of each independent event. Additionally, the probabilities of different events when flipping two coins were explained.
Step-by-step explanation:
The theoretical probability of an event is the expected frequency of occurrence of an event when an experiment is repeated a large number of times. When flipping a fair coin and rolling a fair six-sided die, the sample space for the coin is {H, T} where 'H' stands for heads and 'T' for tails, and the sample space for the die is {1, 2, 3, 4, 5, 6}. Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
For rolling a 1 or a 6, there are 2 favorable outcomes in a sample space of 6 possible outcomes, so P(1 or 6) = 2/6. For flipping tails, there is 1 favorable outcome in a sample space of 2 possible outcomes, so P(tails) = 1/2. To find the probability of both independent events happening together, we multiply the probabilities of each event: P(1 or 6 and tails) = P(1 or 6) * P(tails) = (2/6) * (1/2) = 1/6.
When flipping two fair coins, Sample Space is {HH, HT, TH, TT}. The probability of at most one tail (F) includes {HH, HT, TH} and is 3/4. The probability of two identical faces (G) is 1/2, including {HH, TT}. Lastly, the probability of a head on the first flip followed by either outcome on the second (H) is 1/2, including {HH, HT}.
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