Hallie is trying to win the grand prize on a game show. Should she try her luck by spinning a wheel with 6 equal sections labeled from 1 to 6 and hope she gets a 5, or should she roll two number cubes and hope she gets the same number on both cubes? Explain
Hallie's chances of winning the grand prize are the same with either option.
On the wheel, there is only one section out of six that will win her the grand prize (assuming each section is labeled with a different number). This means she has a 1/6 chance of winning the grand prize by spinning the wheel.
When rolling two number cubes, there are 36 possible outcomes (6 numbers on the first cube multiplied by 6 numbers on the second cube). Out of these possible outcomes, there are 6 outcomes where the same number is rolled on both cubes (1-1, 2-2, 3-3, 4-4, 5-5, 6-6). This means she also has a 1/6 chance of winning the grand prize by rolling two number cubes.
Therefore, it doesn't matter which option Hallie chooses as her chances of winning the grand prize are the same.
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To determine the best strategy for Hallie to win the grand prize, we need to compare the probabilities of success for each option.
Option 1: Spinning a wheel with 6 equal sections labeled from 1 to 6 and hoping to get a 5.
In this case, there is only one section labeled 5 out of the six equal sections. So, the probability of success is 1/6 or approximately 0.1667, which means Hallie has a 16.67% chance of winning the grand prize.
Option 2: Rolling two number cubes and hoping to get the same number on both cubes.
A standard number cube has six sides, labeled with numbers from 1 to 6. Since Hallie needs to get the same number on both cubes, we need to consider the probability of rolling any specific number on the first cube and then the probability of rolling the same number on the second cube.
The probability of rolling any specific number on the first cube is 1/6, as there is an equal chance of rolling any of the six numbers.
When we roll the second cube, we need to get the same number as the first cube. There is a 1 in 6 chance of rolling the same number on the second cube, as there is only one matching number out of the six possible outcomes.
To find the combined probability, we multiply the probabilities of each event. So, the probability of rolling the same number on both cubes is (1/6) * (1/6) = 1/36, which is approximately 0.0278. This means Hallie has a 2.78% chance of winning the grand prize with this option.
Comparing the probabilities, we can see that spinning the wheel and hoping to get a 5 has a higher probability of success (16.67%) compared to rolling two number cubes and hoping to get the same number on both cubes (2.78%). Thus, Hallie should try her luck by spinning the wheel as it offers a higher chance of winning the grand prize.