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 with details .
Both options have an equal probability of winning, which is 1/6. However, rolling two number cubes gives Hallie more chances to win than spinning the wheel. There are six possible outcomes when spinning the wheel, but there are 36 possible outcomes when rolling two number cubes (6 outcomes for the first cube times 6 outcomes for the second cube). Out of those 36 possible outcomes, there are 6 outcomes where the same number appears on both cubes, which gives Hallie a higher chance of winning. Therefore, she should choose to roll two number cubes rather than spin the wheel.
could you show work or this? please and thank youu
Sure! Let's first calculate the probability of winning by spinning the wheel. Since there are 6 equal sections labeled from 1 to 6, the probability of getting a 5 on one spin is 1/6. Therefore:
Probability of winning by spinning the wheel = 1/6
Now let's calculate the probability of winning by rolling two number cubes. Each cube has 6 sides, and each side is numbered from 1 to 6. The probability of getting any number on one cube is 1/6. To calculate the probability of getting the same number on both cubes, we need to multiply the probability of getting any number on one cube by the probability of getting the same number on the other cube. Therefore:
Probability of winning by rolling two number cubes = Probability of getting any number on one cube x Probability of getting the same number on the other cube
Probability of getting any number on one cube = 1/6
Probability of getting the same number on the other cube = 1/6
Probability of winning by rolling two number cubes = (1/6) x (1/6) = 1/36
As we can see, the probability of winning by rolling two number cubes is 1/36, which is higher than the probability of winning by spinning the wheel (1/6). This means that Hallie has a better chance of winning if she chooses to roll two number cubes instead of spinning the wheel.
To determine which option is more likely to help Hallie win the grand prize, let's analyze both scenarios:
1. Spinning a wheel with 6 equal sections:
When Hallie spins the wheel, there are a total of 6 possible outcomes - numbers 1 to 6. Since the wheel sections are equal, the probability of landing on any specific number is 1/6. Therefore, the probability of getting a 5 on the wheel is also 1/6.
2. Rolling two number cubes:
When Hallie rolls two number cubes, each cube has 6 faces, numbered from 1 to 6. In this scenario, Hallie needs to roll the same number on both cubes to win. Let's examine the possible outcomes:
- There are 6 possible outcomes for the first cube (1, 2, 3, 4, 5, 6).
- For each of these outcomes, there is a 1/6 chance of getting the same number on the second cube.
So, for each of the 6 possible outcomes of the first cube, Hallie has a 1/6 chance of getting the same number on the second cube. Therefore, the probability of rolling the same number on both cubes is 1/6 × 1/6 = 1/36.
Comparing the two scenarios:
- The probability of getting a 5 on the wheel is 1/6.
- The probability of rolling the same number on both cubes is 1/36.
Based on probability, it is more likely for Hallie to win the grand prize by spinning the wheel and hoping for a 5, as the probability of getting a 5 is higher (1/6) compared to rolling the same number on both cubes (1/36).