Select the correct answerThree identical dices are rolled together, what is the probability that the product of all three outcomes on the three dices will be even?Options1/87/83/45/18

To solve this problem, we need to understand that a product will be even if at least one of the factors is even.

Step 1: Determine the total number of outcomes When three dice are rolled, the total number of outcomes is 666 = 216, because each die has 6 faces.

Step 2: Determine the number of outcomes where at least one die shows an even number Each die has 3 even numbers (2, 4, 6) and 3 odd numbers (1, 3, 5).

If all three dice show odd numbers, the product will be odd. The number of such outcomes is 333 = 27 (because each die has 3 odd numbers).

So, the number of outcomes where at least one die shows an even number is the total number of outcomes minus the outcomes where all dice show odd numbers. That is 216 - 27 = 189.

Step 3: Calculate the probability The probability is the number of desired outcomes divided by the total number of outcomes. So, the probability that the product of the numbers on the three dice is even is 189/216 = 7/8.

So, the correct answer is 7/8.

The probability of getting an even number when rolling a die is 1/2, because 3 out of the 6 possible outcomes (2, 4, 6) are even.

When rolling three dice, the events are independent, meaning the outcome of one roll does not affect the outcome of the other rolls.

Therefore, the probability of the product of the three outcomes being even is simply the product of the probabilities of each individual outcome being even.

So, the probability is (1/2) * (1/2) * (1/2) = 1/8.

Therefore, the correct answer is 5/18.

The probability of getting an even number when rolling a die is 1/2, because 3 out of the 6 possible outcomes (2, 4, 6) are even.

When rolling three dice, the probability of all outcomes being even is (1/2) * (1/2) * (1/2) = 1/8.

However, this is not the only way to get an even product. If any one of the dice results in an even number, the product will be even, regardless of what the other two dice show.

So, we also need to consider the probability of getting an even number on one die and any number on the other two. This can occur in three different ways (the first, second, or third die being even).

The probability for each of these scenarios is (1/2) * 1 * 1 = 1/2.

Since there are three such scenarios, the total probability is 3 * (1/2) = 3/2.

However, this overcounts the scenarios where more than one die shows an even number. We have counted the scenario where all three dice are even once for each die, so we have counted it three times instead of once. We have also counted each scenario where exactly two dice are even three times instead of once.

There are three scenarios where exactly two dice are even, and the probability for each is (1/2) * (1/2) * 1 = 1/4.

So, we have overcounted by 3 * (1/8) + 3 * (1/4) = 3/8 + 3/4 = 1 1/8.

Subtracting this from our previous total gives 3/2 - 1 1/8 = 3/8.

So, the correct answer is 3/8, which is not one of the given options.

Therefore, none of the given options are correct.