Example Question - rolling a die
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Calculating Probability of Independent Events
In this problem, we're trying to find the probability of two independent events occurring in sequence. First, we need to roll a prime number on a 6-sided die, and second, we need to roll a number less than 4. The prime numbers on a 6-sided die are 2, 3, and 5. Since there are 6 possible outcomes when rolling a die, the probability of rolling a prime number is the number of prime outcomes divided by the total number of possible outcomes. There are 3 prime numbers out of 6 possible outcomes, so the probability of rolling a prime number on a 6-sided die is 3/6, which simplifies to 1/2. For the second event, we want to roll a number less than 4. The numbers less than 4 on a 6-sided die are 1, 2, and 3. There are 3 outcomes that satisfy this condition out of 6 possible outcomes, so the probability of rolling a number less than 4 is also 3/6, which simplifies to 1/2. Since these two events are independent (the outcome of the first roll does not affect the outcome of the second), we can find the combined probability by multiplying the probabilities of each event occurring. So, we multiply the probability of rolling a prime number (1/2) by the probability of rolling a number less than 4 (1/2): Combined probability = (1/2) * (1/2) = 1/4 To express this probability as a percentage, we convert the fraction to a decimal and then to a percentage: Decimal form = 1/4 = 0.25 Percentage form = 0.25 * 100 = 25% Therefore, the probability of rolling a prime number and then rolling a number less than 4, in that order, is 25%.
Calculating Probability of Two Events Occurring in Succession
The question asks for the probability of two events happening in succession: rolling a prime number on a 6-sided die, and then rolling a number less than 4. The prime numbers on a 6-sided die are 2, 3, and 5. Therefore, there are 3 outcomes out of 6 that are prime numbers. The probability of rolling a prime number is therefore 3/6 or 1/2. The numbers less than 4 on a 6-sided die are 1, 2, and 3. So, there are 3 outcomes out of 6 that are less than 4. The probability of rolling a number less than 4 is also 3/6 or 1/2. Since these two events are independent (the outcome of the second roll does not depend on the outcome of the first roll), the total probability of both events occurring in succession is the product of their individual probabilities: (1/2) * (1/2) = 1/4 As a percentage, this is: 1/4 * 100 = 25% Therefore, the probability of rolling a prime number and then rolling a number less than 4 is 25%.
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