Question
Hi! I'm Maddie. I need some help on my homework. Here are three problems I need help with:
A bowl contains only red marbles, blue marbles and green marbles. The probability of selecting a red marble from the bowl is 3/13. The probability of selecting a blue marble from the bowl is 2/5. There are fewer than 100 marbles in the bowl. What is the probability of selecting, at random and without replacement, a green marble and then a red marble from the bowl on the first two selections? Express your answer as a common fraction.
The second one is:
The digits 2, 3, 4, 7, and 8 are each used once in a random order to form a five-digit number. What is the probability that the resulting number is divisible by 4? Express your answer as a common fraction.
And the last one is:
A six-sided die (with numbers 1 through 6) and an eight-sided die (with numbers 1 through 8) are rolled. What is the probability that there is exactly one 6 showing? Express your answer as a common fraction.
Thank you so much if you can help!! It would be amazing if you could help me out.
A bowl contains only red marbles, blue marbles and green marbles. The probability of selecting a red marble from the bowl is 3/13. The probability of selecting a blue marble from the bowl is 2/5. There are fewer than 100 marbles in the bowl. What is the probability of selecting, at random and without replacement, a green marble and then a red marble from the bowl on the first two selections? Express your answer as a common fraction.
The second one is:
The digits 2, 3, 4, 7, and 8 are each used once in a random order to form a five-digit number. What is the probability that the resulting number is divisible by 4? Express your answer as a common fraction.
And the last one is:
A six-sided die (with numbers 1 through 6) and an eight-sided die (with numbers 1 through 8) are rolled. What is the probability that there is exactly one 6 showing? Express your answer as a common fraction.
Thank you so much if you can help!! It would be amazing if you could help me out.
Answers
person
Ha!These are all from AoPS. Anyways,
1) probability of green then red = p(green) * p(red). Take a common multiple of 13 & 5 that's under 100 - I chose 65. The numerators all have to add up to the denominator in the probability. p(red) = 15/65. p(blue) = 26/65. so p(green) has to equal (65 - (15+26))/65 = 24/65. Using cross multiplication, you can find out that 24/65 * 15/65 = 9/104.
1) probability of green then red = p(green) * p(red). Take a common multiple of 13 & 5 that's under 100 - I chose 65. The numerators all have to add up to the denominator in the probability. p(red) = 15/65. p(blue) = 26/65. so p(green) has to equal (65 - (15+26))/65 = 24/65. Using cross multiplication, you can find out that 24/65 * 15/65 = 9/104.
person
2) A number is divisible by 4 if and only if the number formed by its last two digits is divisible by 4. Using the given digits, we find that the only two-digit numbers that are divisible by 4 are 24, 28, 32, 48, 72, and 84.
The probability that the number is divisible by 4 is the number of last-2-digit combinations that make it divisible by four divided by the total ways to choose the last 2 digits.
There are 4 * 5 = 20 ways to choose the last two digits, and they are all equally likely, so the probability that the number is divisible by 4 is 6/20 = 3/10.
The probability that the number is divisible by 4 is the number of last-2-digit combinations that make it divisible by four divided by the total ways to choose the last 2 digits.
There are 4 * 5 = 20 ways to choose the last two digits, and they are all equally likely, so the probability that the number is divisible by 4 is 6/20 = 3/10.
^^theSAMEpersonASabove^^^
3) If there is exactly one 6 showing, then the 6 must be on the six-sided die or the eight-sided die (but not both).
The probability of getting a 6 with the six-sided die is 1/6, and the probability of not rolling a 6 with the eight-sided die is 7/8.
The probability of not getting a 6 with the six-sided die is 5/6, and the probability of rolling a 6 with the eight-sided die is 1/8.
Therefore, the probability that there is exactly one 6 is 1/6 * 7/8 + 5/6 * 1/8 = 7/48 + 5/48 = 12/48 = 1/4
The probability of getting a 6 with the six-sided die is 1/6, and the probability of not rolling a 6 with the eight-sided die is 7/8.
The probability of not getting a 6 with the six-sided die is 5/6, and the probability of rolling a 6 with the eight-sided die is 1/8.
Therefore, the probability that there is exactly one 6 is 1/6 * 7/8 + 5/6 * 1/8 = 7/48 + 5/48 = 12/48 = 1/4
^^theSAMEpersonASabove^^^
So your answers are:
1) 9/104
2) 3/10
3) 1/4
1) 9/104
2) 3/10
3) 1/4