Question

changing probabilities suppose you rolled a twelve - sided number cube instead of a standard six - sided one. how would the probability of these events change? the theoretical probability of landing on a star space on your first roll. the theoretical probability of not landing on a question mark on your first roll. the theoretical probability of reaching the end space on your first roll.

Explanation:

Step1: Recall probability formula

The probability of an event $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

Step2: Analyze probability of landing on star - space

Let's assume there are $n_{s}$ star - spaces. When using a six - sided cube, total number of outcomes $N_1 = 6$. When using a twelve - sided cube, total number of outcomes $N_2=12$. Since $N_2 > N_1$ and number of star - spaces remains the same, the probability $P=\frac{n_{s}}{N}$. As $N$ increases from 6 to 12, the probability of landing on a star space on the first roll decreases.

Step3: Analyze probability of not landing on question - mark

Let's assume there are $n_{q}$ question - mark spaces. The number of non - question - mark spaces is $N - n_{q}$. When $N$ increases from 6 to 12, the proportion of non - question - mark spaces to the total number of spaces changes. Since the number of non - question - mark spaces relative to the total number of spaces decreases as the total number of spaces increases from 6 to 12, the probability of not landing on a question mark on the first roll decreases.

Step4: Analyze probability of reaching end space

Let's assume there is 1 end space. When using a six - sided cube, total number of outcomes is 6. When using a twelve - sided cube, total number of outcomes is 12. Since the number of end spaces (1) remains the same and the total number of outcomes increases from 6 to 12, the probability of reaching the end space on the first roll decreases.

Answer:

The theoretical probability of landing on a star space on your first roll: decreases
The theoretical probability of not landing on a question mark on your first roll: decreases
The theoretical probability of reaching the end space on your first roll: decreases