Determine the following probabilities. Enter your final answers as reduced fractions .
1. Two marbles are chosen at random from a jar of 12 marbles without replacement. There are 8 blue marbles, and 4 red marbles.
What is the probability that the first marble chosen is red and the second marble chosen is red?
2. Two cards are drawn at random from a deck of 52 cards without replacing the first card before choosing the second card.
What is the probability that the first card is a diamond and the second card is a number card that is black ?
explain?
To solve these problems, we need to use the concept of probability and understand how to calculate it.
Probability is a measure of the likelihood of an event happening. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
To calculate the probability of an event, we divide the number of favorable outcomes by the total number of possible outcomes.
Now let's solve the given problems step by step:
1. Two marbles are chosen at random from a jar of 12 marbles without replacement. There are 8 blue marbles and 4 red marbles.
The probability that the first marble chosen is red is 4/12, as there are 4 red marbles out of a total of 12.
After removing one red marble, there are now 3 red marbles and 11 total marbles left in the jar. The probability that the second marble chosen is red is 3/11.
To find the probability that both events occur, we multiply the probabilities together:
Probability of both events = (4/12) * (3/11) = 12/132
The reduced fraction for 12/132 is 1/11.
So, the probability that the first marble chosen is red and the second marble chosen is red is 1/11.
2. Two cards are drawn at random from a deck of 52 cards without replacing the first card before choosing the second card.
The probability that the first card is a diamond is 13/52, as there are 13 diamonds out of a total of 52 cards.
After removing one card, there are now 51 cards left in the deck. The probability that the second card chosen is a number card (2-10) that is black is 20/51.
To find the probability that both events occur, we multiply the probabilities together:
Probability of both events = (13/52) * (20/51) = 260/2652
We can reduce the fraction 260/2652 by dividing both numerator and denominator by their greatest common divisor, which is 4:
Reduced fraction = (260/4) / (2652/4) = 65/663
So, the probability that the first card is a diamond and the second card is a number card that is black is 65/663.