91Ó°ÊÓ

The number of correct identifications can range from 0 (no correct identifications) to 5 (all correct identifications). So, the possible values of \(x\) are \(\{0, 1, 2, 3, 4, 5\}\).

To calculate the probability \(p(x)\) of getting \(x\) correct identifications, we can use the Binomial Probability formula: \[p(x) = \binom{n}{x} p^x (1-p)^{n-x}\] In our case, \(n = 5\) and \(p = \frac{1}{4}\). So, for each possible value of \(x\), we can calculate \(p(x)\) as follows: 1. \(x = 0\): \(p(0) = \binom{5}{0} \left(\frac{1}{4}\right)^0 \left(1 - \frac{1}{4}\right)^{5-0} = 243 \cdot \frac{3^5}{4^5} \approx 0.2373\) 2. \(x = 1\): \(p(1) = \binom{5}{1} \left(\frac{1}{4}\right)^1 \left(1 - \frac{1}{4}\right)^{5-1} = 405 \cdot \frac{3^4}{4^5} \approx 0.3955\) 3. \(x = 2\): \(p(2) = \binom{5}{2} \left(\frac{1}{4}\right)^2 \left(1 - \frac{1}{4}\right)^{5-2} = 270 \cdot \frac{3^3}{4^5} \approx 0.2637\) 4. \(x = 3\): \(p(3) = \binom{5}{3} \left(\frac{1}{4}\right)^3 \left(1 - \frac{1}{4}\right)^{5-3} = 80 \cdot \frac{3^2}{4^5} \approx 0.0889\) 5. \(x = 4\): \(p(4) = \binom{5}{4} \left(\frac{1}{4}\right)^4 \left(1 - \frac{1}{4}\right)^{5-4} = 15 \cdot \frac{3^1}{4^5} \approx 0.0146\) 6. \(x = 5\): \(p(5) = \binom{5}{5} \left(\frac{1}{4}\right)^5 \left(1 - \frac{1}{4}\right)^{5-5}= 1 \cdot \frac{1}{4^5} \approx 0.0010\) Now, let's display the possible \(x\) values and the corresponding \(p(x)\) values in a table: | x | p(x) | |---|---------| | 0 | 0.2373 | | 1 | 0.3955 | | 2 | 0.2637 | | 3 | 0.0889 | | 4 | 0.0146 | | 5 | 0.0010 |

To construct a histogram displaying the probability distribution of \(x\), we can plot the \(x\) values on the horizontal axis and the corresponding \(p(x)\) values on the vertical axis. Each bar in the histogram represents an \(x\) value, and its height is equal to the value of \(p(x)\). The histogram will look like the following: ``` 1.0 ┼ │ * 0.9 ┼ │ 0.8 ┼ │ 0.7 ┼ │ * 0.6 ┼ │ 0.5 ┼ │ * 0.4 ┼ │ 0.3 ┼ * │ 0.2 ┼ │ * 0.1 ┼ │ * 0.0 ┼ ├───┼───┼───┼───┼───┼───┼───┼───┼ 0 1 2 3 4 5 x ``` In this histogram, we can see that the probability of getting 1 correct identification (\(x=1\)) is the highest, while the probabilities of getting all the identifications correct (\(x=5\)) or getting none of them correct (\(x=0\)) are the lowest.