Question

1. at laras school, 75% of the student body is male and 35% of the students walk to school. assume that these two events are independent. if a student from laras school is selected at random, determine the following probabilities. a. create a probability area model or tree diagram to represent this situation. b. p(student is female) c. p(student is male and does not walk to school) d. p(student is female or does not walk to school) e. identify the sample space in parts (c) and (d) above as a \union\ or an \intersection\.

Explanation:

Step1: Define probabilities

Let $P(M)=0.75$ (probability of being male), so $P(F)=1 - P(M)=1 - 0.75 = 0.25$ (probability of being female). Let $P(W)=0.35$ (probability of walking to school), then $P(\overline{W})=1 - P(W)=1 - 0.35 = 0.65$ (probability of not - walking to school). Since the two events (gender and walking to school) are independent, if $A$ and $B$ are independent events, $P(A\cap B)=P(A)\times P(B)$.

Step2: Calculate $P(F)$

We know that the sum of the probabilities of all genders is 1. Since $P(M) = 0.75$, then $P(F)=1 - 0.75=0.25$.

Step3: Calculate $P(M\cap\overline{W})$

Using the formula for independent events $P(A\cap B)=P(A)\times P(B)$, we substitute $A = M$ and $B=\overline{W}$. So $P(M\cap\overline{W})=P(M)\times P(\overline{W})=0.75\times0.65 = 0.4875$.

Step4: Calculate $P(F\cup\overline{W})$

Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here $A = F$ and $B=\overline{W}$. Since $F$ and $\overline{W}$ are independent, $P(F\cap\overline{W})=P(F)\times P(\overline{W})=0.25\times0.65 = 0.1625$. Then $P(F\cup\overline{W})=P(F)+P(\overline{W})-P(F\cap\overline{W})=0.25 + 0.65-0.1625=0.7375$.

Step5: Identify sample - space type

For part (c), $P(M\cap\overline{W})$ represents an intersection (the event of being male and not walking to school is the intersection of the two events “being male” and “not walking to school”). For part (d), $P(F\cup\overline{W})$ represents a union (the event of being female or not walking to school is the union of the two events “being female” and “not walking to school”).

Answer:

b. $0.25$
c. $0.4875$
d. $0.7375$
e. c: intersection; d: union