A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 40% choose jumping. Fifteen percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that one student joins the dressage team and the other student does not?
A)0.2275
B)0.1925
C)0.2475
D)0.1525

Question

Quizplus · Accepted Answer

We need to find the probability of selecting one student for the dressage team (45% of the total students) and the other student for either jumping or Western riding (the remaining 55% of the total students).

Let's use a Venn diagram to help solve the problem:

[asy] size(8cm); label("Western", (1.5,60)); label("Dressage", (50,60)); label("Jumping", (98.5,60)); draw(Circle((25,30),20)); draw(Circle((73,30),20)); draw(Circle((49,56),20)); label("10%", (37.5,35)); label("15%", (61.5,35)); label("x", (37.5,20)); label("y", (61.5,20)); label("35%", (1.5,40)); label("45%", (50,40)); label("40%", (98.5,40));  draw((23.5,33)--(25,30)--(26.5,33)); draw((71.5,33)--(73,30)--(74.5,33)); [/asy]

According to the Venn diagram, we know that: 
- 10% of the students choose Western and dressage, so the area labeled x equals 10%.
- 15% of the students choose both dressage and jumping, so the area labeled y equals 15%.
- No one chooses Western and jumping, so the area outside all three circles is 35% + 40% = 75%.

To find the value of x and y, we can use the fact that the total percentage adds up to 100%. We set up two equations:

x + 10% + y = 45% 
x + 15% + y = 55%

Solving for x and y, we get: 
x = 20% 
y = 20%

Now we can find the probability of selecting one student for the dressage team and the other for either jumping or Western riding.

Probability = P(dressage) * P(not dressage) + P(not dressage) * P(dressage) 
Probability = 0.45 * 0.55 + 0.55 * 0.45 
Probability = 0.2475

Therefore, the probability that one student joins the dressage team and the other student does not is 0.2475 or approximately 24.75%.

A Group of Freshmen at a Local University Consider Joining