5.3b Statistics - Probability Of Complements And Compound Events

Explanation
When a single six-sided die is rolled, there are a total of six possible outcomes, which are the numbers 1, 2, 3, 4, 5, and 6. The probability of rolling a 4 is 1 out of 6, since there is only one 4 on the die. Therefore, the probability of not rolling a 4 is 5 out of 6, because there are five other numbers on the die. So, the correct answer is 5/6.

Explanation
When two six-sided dice are rolled, there are 36 possible outcomes. To find the probability that the sum of a roll is not 5, we need to count the number of outcomes where the sum is not 5. From the given outcomes, we can see that there are 4 outcomes where the sum is 5: (1,4), (2,3), (3,2), and (4,1). Therefore, the number of outcomes where the sum is not 5 is 36 - 4 = 32. The probability is then calculated by dividing the number of favorable outcomes (32) by the total number of possible outcomes (36), which gives us 32/36.

Explanation
When two six-sided dice are rolled, there are a total of 36 possible outcomes, as each die has 6 possible outcomes and there are 6 possibilities for the second die for each outcome of the first die. The question asks for the probability that the sum of the roll is not 12. Since there is only one outcome where the sum is 12 (6,6), and there are 36 possible outcomes in total, the probability of not rolling a sum of 12 is 35/36.

Explanation
When two six-sided dice are rolled, there are 36 possible outcomes because each die has 6 possible outcomes, and the total number of outcomes is found by multiplying the number of outcomes for each die (6 x 6 = 36). In this case, the question asks for the probability that the sum of the roll is 12. There is only one outcome where the sum is 12, which is when both dice show a 6. Therefore, the probability is 1 out of 36, or 1/36.

Explanation
When two six-sided dice are rolled, each die has 6 possible outcomes. Since there are 2 dice, the total number of possible outcomes is 6 * 6 = 36.

To find the probability that the sum of a roll is not 7, we need to count the number of outcomes where the sum is not 7.

Out of the 36 possible outcomes, there are 6 outcomes where the sum is 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

Therefore, the number of outcomes where the sum is not 7 is 36 - 6 = 30.

The probability of getting a sum that is not 7 is the number of favorable outcomes (30) divided by the total number of possible outcomes (36), which simplifies to 30/36 or 5/6.

Explanation
The probability that the student selected for the interview won an award for either English or mathematics can be found by adding the number of students who won awards for English and the number of students who won awards for mathematics, and then subtracting the number of students who won awards for both. This gives us a total of 126 students. Since there are 162 students in total, the probability is 126/162, which simplifies to 7/9.

Explanation
The probability of selecting a king from a standard 52-card deck is 4/52 since there are 4 kings in the deck. The probability of selecting a spade is 13/52 since there are 13 spades in the deck. To find the probability of selecting either a king or a spade, we add the probabilities together: 4/52 + 13/52 = 16/52. Therefore, the probability that the card selected is either a king or a spade is 16/52 or 4/13.

Explanation
The probability of selecting a queen from a standard 52-card deck is 4/52, since there are 4 queens in the deck. Similarly, the probability of selecting a 3 is also 4/52. However, since the question asks for the probability of selecting either a queen or a 3, we need to add these probabilities together. Therefore, the probability is 4/52 + 4/52 = 8/52, which can be simplified to 2/13.

Explanation
The probability of an event occurring is calculated by multiplying the probabilities of the individual events. In this case, the probability of rain on both Saturday and Sunday is calculated by multiplying the probability of rain on Saturday (20%) with the probability of rain on Sunday (40%). Therefore, the probability that it will rain on both Saturday and Sunday is 0.2 * 0.4 = 0.08, which is equivalent to 8%.

Explanation
The probability of an event occurring is calculated by multiplying the probabilities of each independent event together. In this case, the probability of rain on both Saturday and Sunday would be 40% (the probability of rain on Saturday) multiplied by 70% (the probability of rain on Sunday), which equals 28%. However, the question asks for the probability that it will rain on both Saturday and Saturday, which is not possible. Therefore, the correct answer is 0%.