Question

a school conducts a recipe contest in which 1 student gets first place, 1 student gets second place, 1 student gets third place, and 10 students tie for fourth place. fifty students enter the contest. if ( x ) represents the distribution of prizes, which graph represents the probability distribution for a randomly selected student? four probability distribution bar graphs are shown, with categories no prize, 1st, 2nd, 3rd, 4th on the x - axis and ( p_x(x) ) on the y - axis. the first graph has y - axis from 0 to 0.8, the second from 0 to 40, the third from 0 to 0.9, and the fourth from 0 to 0.8, with different bar heights for each category.

Explanation:

Step1: Calculate number of students with no prize

Total students = 50. Students with prizes: 1 (1st) + 1 (2nd) + 1 (3rd) + 10 (4th) = 13. So no - prize students = 50 - 13 = 37.

Step2: Calculate probabilities

• Probability of no prize: $P(\text{no prize})=\frac{37}{50}=0.74$
• Probability of 1st place: $P(1st)=\frac{1}{50}=0.02$
• Probability of 2nd place: $P(2nd)=\frac{1}{50}=0.02$
• Probability of 3rd place: $P(3rd)=\frac{1}{50}=0.02$
• Probability of 4th place: $P(4th)=\frac{10}{50}=0.2$

We need to find the graph where the bar for "No prize" has a height around 0.74, "1st", "2nd", "3rd" have small heights (around 0.02) and "4th" has a height of 0.2. Looking at the graphs, the first graph (top - left) has a "No prize" bar around 0.7 (close to 0.74), "1st", "2nd", "3rd" with small heights and "4th" around 0.2. The second graph has a y - axis with values up to 40 (not probability as probability is between 0 and 1), the third graph has "No prize" around 0.8 (higher than 0.74) and the fourth graph has "4th" bar height not matching 0.2. So the correct graph is the top - left probability distribution graph.

Answer:

The top - left "Probability Distribution" graph (with $P_X(x)$ on y - axis from 0 to 0.8, "No prize" bar around 0.7, small bars for 1st, 2nd, 3rd and a bar around 0.2 for 4th).