To tackle this problem, let's go through it step-by-step.
### Step 1: Determine the probability of rolling a '2'
First, let's establish the probability of rolling a '2' on this specially designed six-sided number cube. The cube has face values: 1, 1, 2, 2, 5, 6.
There are two '2's out of six faces. Therefore, the probability [tex]\( p \)[/tex] of rolling a '2' is:
[tex]\[ p = \frac{\text{Number of '2's}}{\text{Total number of faces}} = \frac{2}{6} = \frac{1}{3} \approx 0.3333 \][/tex]
### Step 2: Calculate the expected number of '2's in 50 rolls
Next, determine the expected number (mean) of '2's in 50 rolls. This mean is given by the product of the total number of rolls and the probability of rolling a '2':
[tex]\[ \text{Mean} = \text{Total rolls} \times p = 50 \times \frac{1}{3} \approx 16.67 \][/tex]
### Step 3: Calculate the standard deviation
For a binomial distribution, the standard deviation [tex]\( \sigma \)[/tex] can be calculated using the formula:
[tex]\[ \sigma = \sqrt{n \times p \times (1 - p)} \][/tex]
Here, [tex]\( n \)[/tex] is the number of trials (rolls), and [tex]\( p \)[/tex] is the probability of a '2':
[tex]\[ \sigma = \sqrt{50 \times \frac{1}{3} \times \left( 1 - \frac{1}{3} \right)} \approx 3.33 \][/tex]
### Step 4: Set up the threshold value
We are interested in finding the probability that fewer than 30% of the rolls result in a '2'. In 50 rolls, 30% is:
[tex]\[ \text{Threshold} = 0.30 \times 50 = 15 \][/tex]
### Step 5: Calculate the z-score
Next, determine the z-score corresponding to this threshold value using the formula:
[tex]\[ z = \frac{\text{Threshold} - \text{Mean}}{\sigma} \][/tex]
Plugging in the values we have:
[tex]\[ z = \frac{15 - 16.67}{3.33} \approx -0.5 \][/tex]
### Step 6: Find the desired probability using the z-score
Finally, use the z-table to find the probability corresponding to a z-score of -0.5.
From the z-table, a z-score of -0.5 corresponds to a cumulative probability of approximately 0.3085.
Thus, the probability that fewer than 30% of the rolls result in a '2' is approximately [tex]\( 0.309 \)[/tex] or [tex]\( 30.85\% \)[/tex].
So, the detailed answer is:
1. Probability of rolling a '2' is 0.3333.
2. The expected number of '2's in 50 rolls is approximately 16.67.
3. The standard deviation of the distribution is approximately 3.33.
4. The threshold for fewer than 30% of 50 rolls is 15.
5. The z-score for this threshold is approximately -0.5.
6. The cumulative probability corresponding to a z-score of -0.5 is approximately 0.3085.
Hence, the probability that fewer than 30% of the rolls result in a '2' is about 0.3085 or 30.85%.
