Question
Part B Constructed Response Complete all items in the space provided. Show all workings. 1. A bag contains 4 red marbles and some green marbles. The odds of choosing a red marble from the bag are 2:3 (a) How many green marbles are in the bag? (2 marks) (b) What is the probability of choosing a red marble from the bag? Give your answer as a fraction in lowest terms. (2 marks) 2. If a four-digit number is generated at random from the digits 2,3,5 and 7 (without repeating a digit)what is the probability that the four-digit number will be: (a) An even number? Give your answer as a fraction in lowest terms. (3 marks) (b) An odd number? Give your answer as a fraction in lowest terms (3 marks) 3. To win a prize at a local radio station, a contestant needs to spell out the word SASKATOON with letter tiles. If the tiles are all mixed up and face down, what is the probability that the contestant will win the prize? Give your answer as a fraction in lowest terms (3 marks)
Solution
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Answer
### 1. (a) 6 green marbles; (b) \frac{2}{5}
### 2. (a) \frac{1}{4}; (b) \frac{3}{4}
### 3. \frac{1}{90720}
### 2. (a) \frac{1}{4}; (b) \frac{3}{4}
### 3. \frac{1}{90720}
Explain
## Step 1: Finding the Number of Green Marbles
### The odds of choosing a red marble are given as 2:3. This means for every 2 red marbles, there are 3 green marbles. Since there are 4 red marbles, we can set up a proportion: \frac{2}{3} = \frac{4}{x}, where x is the number of green marbles. Solving for x, we get x = \frac{4 \times 3}{2} = 6.
## Step 2: Calculating the Probability of Choosing a Red Marble
### The total number of marbles is 4 \text{ (red)} + 6 \text{ (green)} = 10. The probability of choosing a red marble is the number of red marbles divided by the total number of marbles: \frac{4}{10}. Simplifying this fraction gives \frac{2}{5}.
## Step 3: Calculating the Total Number of Four-Digit Numbers
### There are 4 choices for the first digit, 3 choices for the second, 2 for the third, and 1 for the last. The total number of four-digit numbers is 4 \times 3 \times 2 \times 1 = 24.
## Step 4: Calculating the Probability of an Even Number
### For the number to be even, the last digit must be 2. This leaves 3 choices for the first digit, 2 for the second, and 1 for the third. The number of even numbers is 3 \times 2 \times 1 \times 1 = 6. The probability of an even number is \frac{6}{24} = \frac{1}{4}.
## Step 5: Calculating the Probability of an Odd Number
### Since there are only even and odd numbers, the probability of an odd number is 1 - \text{probability of an even number} = 1 - \frac{1}{4} = \frac{3}{4}. Alternatively, for the number to be odd, the last digit must be 3, 5, or 7. There are 3 options. This leaves 3 choices for the first digit, 2 for the second, and 1 for the third. The number of odd numbers is 3 \times 2 \times 1 \times 3 = 18. The probability is \frac{18}{24} = \frac{3}{4}.
## Step 6: Calculating the Probability of Spelling SASKATOON
### The word SASKATOON has 9 letters. There are two S's, two O's, and one each of A, K, T, N. The total number of arrangements is \frac{9!}{2!2!} = 90720. Since there is only one correct arrangement to win, the probability is \frac{1}{90720}.
### The odds of choosing a red marble are given as 2:3. This means for every 2 red marbles, there are 3 green marbles. Since there are 4 red marbles, we can set up a proportion: \frac{2}{3} = \frac{4}{x}, where x is the number of green marbles. Solving for x, we get x = \frac{4 \times 3}{2} = 6.
## Step 2: Calculating the Probability of Choosing a Red Marble
### The total number of marbles is 4 \text{ (red)} + 6 \text{ (green)} = 10. The probability of choosing a red marble is the number of red marbles divided by the total number of marbles: \frac{4}{10}. Simplifying this fraction gives \frac{2}{5}.
## Step 3: Calculating the Total Number of Four-Digit Numbers
### There are 4 choices for the first digit, 3 choices for the second, 2 for the third, and 1 for the last. The total number of four-digit numbers is 4 \times 3 \times 2 \times 1 = 24.
## Step 4: Calculating the Probability of an Even Number
### For the number to be even, the last digit must be 2. This leaves 3 choices for the first digit, 2 for the second, and 1 for the third. The number of even numbers is 3 \times 2 \times 1 \times 1 = 6. The probability of an even number is \frac{6}{24} = \frac{1}{4}.
## Step 5: Calculating the Probability of an Odd Number
### Since there are only even and odd numbers, the probability of an odd number is 1 - \text{probability of an even number} = 1 - \frac{1}{4} = \frac{3}{4}. Alternatively, for the number to be odd, the last digit must be 3, 5, or 7. There are 3 options. This leaves 3 choices for the first digit, 2 for the second, and 1 for the third. The number of odd numbers is 3 \times 2 \times 1 \times 3 = 18. The probability is \frac{18}{24} = \frac{3}{4}.
## Step 6: Calculating the Probability of Spelling SASKATOON
### The word SASKATOON has 9 letters. There are two S's, two O's, and one each of A, K, T, N. The total number of arrangements is \frac{9!}{2!2!} = 90720. Since there is only one correct arrangement to win, the probability is \frac{1}{90720}.
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