Problem: A six-sided die is thrown once. Find the probability of getting a number which is a multiple of 3.

A six-sided die is thrown once. Find the probability of getting a number which is a multiple of 3.

• Step 1: Identify the total possible outcomes when rolling a six-sided die. The sample space is $\{1, 2, 3, 4, 5, 6\}$. Total number of outcomes = 6.
• Step 2: Identify the favorable outcomes (multiples of 3). The multiples of 3 in the sample space are $\{3, 6\}$. Number of favorable outcomes = 2.
• Step 3: Calculate the probability. $$P(\text{multiple of 3}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{6} = \frac{1}{3}$$ The probability of getting a multiple of 3 is $\boxed{\frac{1}{3}}$.

A fair die is rolled once. Find the probability of a 4 showing up.

• Step 1: Identify the total possible outcomes when rolling a fair die. The sample space is $\{1, 2, 3, 4, 5, 6\}$. Total number of outcomes = 6.
• Step 2: Identify the favorable outcome (a 4 showing up). The favorable outcome is $\{4\}$. Number of favorable outcomes = 1.
• Step 3: Calculate the probability. $$P(\text{4 showing up}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{6}$$ The probability of a 4 showing up is $\boxed{\frac{1}{6}}$.

A letter is chosen at random from the letters of the word "EXCELLENT". What is the probability that the letter "E" is chosen?

• Step 1: Count the total number of letters in the word "EXCELLENT". The letters are E, X, C, E, L, L, E, N, T. Total number of letters = 9.
• Step 2: Count the number of times the letter "E" appears. The letter "E" appears 3 times. Number of favorable outcomes = 3.
• Step 3: Calculate the probability. $$P(\text{choosing E}) = \frac{\text{Number of E's}}{\text{Total number of letters}} = \frac{3}{9} = \frac{1}{3}$$ The probability that the letter "E" is chosen is $\boxed{\frac{1}{3}}$.

In a bag there are 4 blue marbles, 3 red marbles and 2 white marbles. What is the probability of picking at random a red marble?

• Step 1: Calculate the total number of marbles in the bag. Total marbles = $4 (\text{blue}) + 3 (\text{red}) + 2 (\text{white}) = 9$ marbles.
• Step 2: Identify the number of red marbles. Number of red marbles = 3.
• Step 3: Calculate the probability of picking a red marble. $$P(\text{picking a red marble}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{3}{9} = \frac{1}{3}$$ The probability of picking a red marble is $\boxed{\frac{1}{3}}$.

The probability that a boy will be late for school on any particular day is $x$. Find, in terms of $x$, the probability that he will not be late for school.

• Step 1: Understand the relationship between an event and its complement. The probability of an event happening plus the probability of it not happening is equal to 1. $P(\text{event}) + P(\text{not event}) = 1$.
• Step 2: Apply this to the given problem. Let $P(\text{late})$ be the probability that the boy is late, which is given as $x$. Let $P(\text{not late})$ be the probability that the boy is not late. $$P(\text{not late}) = 1 - P(\text{late})$$ $$P(\text{not late}) = 1 - x$$ The probability that he will not be late for school is $\boxed{1 - x}$.

The probability of Kasukulu waking up late is 0.3. What is the probability that she will wake up early?

• Step 1: Assume that "waking up early" is the complement of "waking up late". The sum of the probabilities of these two events must be 1. $P(\text{late}) + P(\text{early}) = 1$.
• Step 2: Use the given probability to find the unknown probability. Given $P(\text{late}) = 0.3$. $$P(\text{early}) = 1 - P(\text{late})$$ $$P(\text{early}) = 1 - 0.3$$ $$P(\text{early}) = 0.7$$ The probability that she will wake up early is $\boxed{0.7}$.