A spinner has six equally spaced sections numbered 1 to 6. You spin the spinner four times.
In how many ways could the spinner result in an even number or the same colour on all four spins?
To find the number of ways the spinner could result in an even number or the same color on all four spins, we need to consider two cases:
Case 1: Even number on all four spins.
There are three even numbers on the spinner (2, 4, and 6). For each spin, we have 3 possible outcomes (even numbers), so the total number of outcomes for four spins is: 3 x 3 x 3 x 3 = 81.
Case 2: Same color on all four spins.
There are two colors on the spinner. For each spin, we have 2 possible outcomes (same color or different color). So the total number of outcomes for four spins is: 2 x 2 x 2 x 2 = 16.
However, we need to subtract the cases where all four spins result in an even number and the same color. These cases have been counted twice in the above cases.
Case 3: Even number and same color on all four spins.
There is only one even number (6) that can be of the same color, and there is only one such color. So the total number of outcomes for this case is: 1.
Therefore, the total number of ways the spinner could result in an even number or the same color on all four spins is:
81 + 16 - 1 = 96.
To find the number of ways the spinner could result in an even number or the same color on all four spins, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
First, let's find the favorable outcomes for each condition separately.
1. Even number on all four spins:
Since there are three even numbers on the spinner (2, 4, 6), the probability of getting an even number on one spin is 3/6 = 1/2. Since we want an even number on all four spins, the probability multiplies 4 times:
(1/2) * (1/2) * (1/2) * (1/2) = 1/16
2. Same color on all four spins:
Since there are two colors on the spinner, the probability of getting the same color on one spin is 2/6 = 1/3. Like before, we multiply this probability 4 times:
(1/3) * (1/3) * (1/3) * (1/3) = 1/81
Now, let's find the total number of possible outcomes:
Since there are 6 equally spaced sections on the spinner, the total number of favorable outcomes for one spin is 6. Therefore, the total number of possible outcomes for four spins is 6 * 6 * 6 * 6 = 6^4 = 1296.
To find the number of ways that the spinner could result in an even number or the same color on all four spins, we need to sum the favorable outcomes for each condition:
1/16 + 1/81 = 97/1296
Therefore, there are 97 ways that the spinner could result in an even number or the same color on all four spins.
even: 3*5!
assuming 6 colors, then same color: 6