You are a contestant on a game show and have the chance to win a car by rolling 5 dice. Each die has three sides that say CAR and three sides that are blank. In order for you to win the car you CAR on all five dice. You have three chanat ces to roll the dice. On each roll you can keep any dice that come up CAR and re-roll only the dice that come up blank. Who benefits from this game more: the contestant or the game show owner?

To determine who benefits more from the game, let's analyze the probabilities.

There are two scenarios to consider: the contestant winning the car and the contestant not winning the car.

Scenario 1: The contestant wins the car:
- The probability of rolling CAR on a single die is 3/6 = 1/2.
- The probability of rolling CAR on all five dice in a single roll is (1/2) ^ 5 = 1/32.
- Since the contestant has three chances to roll the dice, the probability of winning the car is 1 - (31/32) ^ 3 = 93.71%.

Scenario 2: The contestant does not win the car:
- The probability of not rolling CAR on a single die is 1 - (1/2) = 1/2.
- The probability of not rolling CAR on all five dice in a single roll is (1/2) ^ 5 = 1/32.
- The probability of the contestant not winning the car in three rolls is (31/32) ^ 3 = 71.49%.

Comparing the probabilities of winning and not winning the car, we can conclude that the contestant benefits more from this game. With a probability of 93.71%, the contestant has a higher chance of winning the car compared to the game show owner's probability of 71.49% of the contestant not winning the car.

To determine who benefits more in this game, let's analyze the probabilities involved for both the contestant and the game show owner.

The contestant has three chances to roll the dice. On each roll, there are three possible outcomes for each die: CAR, blank, or already kept CAR. Let's break down the probabilities for each roll:

First roll:
- Probability of getting CAR on all five dice: (3/6)^5 = 0.0772 (approximately 7.72%)
- Probability of not getting CAR on all five dice: 1 - 0.0772 = 0.9228 (approximately 92.28%)

Second roll:
Assuming the contestant kept any CAR dice from the first roll (if any), they would re-roll the remaining blank dice. The probabilities for this roll depend on the number of blank dice left, which could be 0 to 5. Let's consider the worst-case scenario of all dice being blank after the first roll.

- Probability of getting CAR on all five dice after the second roll: (3/6)^5 = 0.0772 (approximately 7.72%)
- Probability of not getting CAR on all five dice after the second roll: 1 - 0.0772 = 0.9228 (approximately 92.28%)

Third roll:
Similar to the second roll, the probabilities depend on the number of blank dice left after the second roll. Again, let's consider the worst-case scenario of all dice being blank after the second roll.

- Probability of getting CAR on all five dice after the third roll: (3/6)^5 = 0.0772 (approximately 7.72%)
- Probability of not getting CAR on all five dice after the third roll: 1 - 0.0772 = 0.9228 (approximately 92.28%)

Now, let's consider the game show owner's perspective. If the contestant wins the car, the game show owner incurs the cost of providing the car. On the other hand, if the contestant fails to get CAR on all five dice after three rolls, the game show owner does not have to provide the car.

Considering the probabilities of winning and losing for both the contestant and game show owner, it is evident that the game show owner benefits more from this scenario. The probabilities heavily favor the game show owner, with the contestant having only about a 7.72% chance to win the car on each roll. Therefore, the game show owner has a higher probability of not having to provide the car, making the game more profitable for them.