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A number cube is rolled twice. Then, find P(not 2 if even). In its simplest form, express the probability as a fraction.
To solve this problem, we need to break it down into smaller parts and use a combination of probability and mathematical reasoning.
Step 1: Determine the sample space
The sample space is the set of all possible outcomes of rolling a number cube twice. Each roll can result in one of six possible outcomes (1, 2, 3, 4, 5, or 6). Since there are two rolls, there are 6 x 6 = 36 possible outcomes in the sample space.
Step 2: Determine the event of interest
We are interested in the event of rolling an even number that is not 2 on both rolls. To do this, we first need to identify all the even numbers in the sample space. These are 2, 4, and 6. Since we want to exclude the outcome of rolling a 2, we can remove that from the list. So, the even numbers we are interested in are 4 and 6. We need to calculate the probability of rolling either 4 or 6 on both rolls.
Step 3: Calculate the probability of the event of interest
To calculate the probability of rolling either 4 or 6 on both rolls of the number cube, we need to use the multiplication rule of probability. This rule states that the probability of two independent events occurring together is the product of their individual probabilities. Since the rolls of the number cube are independent (i.e., the first roll does not affect the second roll), we can use this rule.
The probability of rolling a 4 on the first roll is 1/6, since there is one outcome that gives 4 out of the six possible outcomes. The probability of rolling a 4 on the second roll is also 1/6, for the same reason. So, the probability of rolling a 4 on both rolls is (1/6) x (1/6) = 1/36.
The probability of rolling a 6 on the first roll is also 1/6, since there is one outcome that gives 6 out of the six possible outcomes. The probability of rolling a 6 on the second roll is also 1/6, for the same reason. So, the probability of rolling a 6 on both rolls is (1/6) x (1/6) = 1/36.
To get the total probability of rolling either 4 or 6 on both rolls, we need to add these two probabilities together: 1/36 + 1/36 = 2/36. Simplifying this fraction, we get 1/18.
Step 4: Determine the complement event and calculate its probability
The complement event is the event of rolling a 2 or an odd number on either roll. We need to determine the probability of this event and then subtract it from 1 to get the probability of the event of interest (rolling an even number that is not 2 on both rolls).
To calculate the probability of rolling a 2 on either roll, we can use the addition rule of probability. This rule states that the probability of two mutually exclusive events occurring is the sum of their individual probabilities. Since rolling a 2 and rolling an odd number are mutually exclusive (i.e., they cannot occur at the same time), we can use this rule.
The probability of rolling a 2 on the first roll is 1/6, since there is one outcome that gives 2 out of the six possible outcomes. The probability of rolling an odd number on the first roll is 3/6, since there are three outcomes that give odd numbers out of the six possible outcomes. So, the probability of rolling a 2 or an odd number on the first roll is (1/6) + (3/6) = 4/6.
The same probabilities apply for the second roll. So, the probability of rolling a 2 or an odd number on either roll is (4/6) x (4/6) = 16/36.
To get the probability of the complement event (rolling an even number that is not 2 on both rolls), we need to subtract the probability of rolling either a 2 or an odd number on either roll from 1. So, the probability of the complement event is 1 - (16/36) = 20/36. Simplifying this fraction, we get 5/9.
Step 5: Express the probability of the event of interest as a fraction
Now that we have determined the probability of the complement event, we can subtract it from 1 to get the probability of the event of interest (rolling an even number that is not 2 on both rolls). So, the probability of rolling an even number that is not 2 on both rolls is 1 - (5/9) = 4/9.
Therefore, P(not 2 if even) = 4/9.