Question Video: Calculating a Probability with Replacement Mathematics

A jar of marbles contains 4 blue marbles, 5 red marbles, 1 green marble, and 2 black marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. Find the probability that the first is blue and the second is red.

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Video Transcript

A jar of marbles contains four blue marbles, five red marbles, one green marble, and two black marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. Find the probability that the first is blue and the second is red.

One of the key parts to this question is the fact that the marble is replaced. This means that we are dealing with independent events. The first marble does not impact the selection of the second marble. This is because the total number of marbles in the jar will remain constant. Every time that a marble is selected from the jar, there will be a total of 12 marbles to choose from. We know that when dealing with independent events, the probability of event A and event B occurring is equal to the probability of A multiplied by the probability of B. This is known as the intersection.

In this question, we will let event A be the probability of selecting a blue marble. Event B is the probability of selecting a red marble. We can write probability as a fraction, where our numerator is the number of successful outcomes and the denominator is the number of possible outcomes for any random event. In this case, the top number or numerator will be the number of marbles of the color we want, and the denominator will be the total number of marbles. There are four blue marbles. Therefore, the probability of event A is four out of 12 or four twelfths. There are five red marbles. Therefore, the probability of event B, selecting a red marble, is five out of 12 or five twelfths.

Before multiplying these fractions, we notice that the first fraction can be canceled. Both the numerator and denominator are divisible by four, so four twelfths simplifies to one-third. We can then multiply the numerators and, separately, the denominators. One multiplied by five is equal to five, and three multiplied by 12 is 36. The probability that the first marble selected is blue and the second is red is five out of 36.

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