SOLUTION: There are 90 players in a tennis club. Of these, 23 are juniors, the rest are seniors. 34 of the seniors and 10 of the juniors are male. There are 8 juniors who are left-handed,

Algebra ->  Probability-and-statistics -> SOLUTION: There are 90 players in a tennis club. Of these, 23 are juniors, the rest are seniors. 34 of the seniors and 10 of the juniors are male. There are 8 juniors who are left-handed,       Log On


   

Question 1194339: There are 90 players in a tennis club. Of these, 23 are juniors, the rest
are seniors. 34 of the seniors and 10 of the juniors are male. There are
8 juniors who are left-handed, 5 of whom are male. There are 18 lefthanded players in total, 4 of whom are female seniors.
(i) Represent this information in a Venn diagram.
(ii) What is the probability that:
(a) a male player selected at random is left-handed?
(b) a left-handed player selected at random is a female junior?
(c) a player selected at random is either a junior or a female?
(d) a player selected at random is right-handed?
(e) a right-handed player selected at random is not a junior?
(f) a right-handed female player selected at random is a junior?

Answer by proyaop(69) About Me  (Show Source):
You can put this solution on YOUR website!
Certainly, let's break down this problem step-by-step.
**i) Venn Diagram**
* **Sets:**
* J: Set of Juniors
* S: Set of Seniors
* M: Set of Males
* L: Set of Left-handed players
* **Populate the Venn Diagram:**
* **Total Players:** 90
* **Juniors (J):** 23
* **Male Juniors (J ∩ M):** 10
* **Female Juniors (J ∩ M'):** 23 - 10 = 13
* **Left-handed Juniors (J ∩ L):** 8
* **Left-handed Female Juniors (J ∩ L ∩ M'):** 8 - 5 = 3
* **Seniors (S):** 90 - 23 = 67
* **Male Seniors (S ∩ M):** 34
* **Female Seniors (S ∩ M'):** 67 - 34 = 33
* **Left-handed Seniors (S ∩ L):** 18 - 8 = 10
* **Left-handed Female Seniors (S ∩ L ∩ M'):** 4
**ii) Calculate Probabilities**
**(a) Probability that a male player selected at random is left-handed:**
* P(L | M) = (Number of left-handed males) / (Total number of males)
* P(L | M) = (10 + 5) / (10 + 34)
* P(L | M) = 15 / 44
* P(L | M) ≈ 0.3409
**(b) Probability that a left-handed player selected at random is a female junior:**
* P(J ∩ M' | L) = (Number of left-handed female juniors) / (Total number of left-handed players)
* P(J ∩ M' | L) = 3 / 18
* P(J ∩ M' | L) = 1/6
* P(J ∩ M' | L) ≈ 0.1667
**(c) Probability that a player selected at random is either a junior or a female:**
* P(J ∪ M') = P(J) + P(M') - P(J ∩ M')
* P(J) = 23/90
* P(M') = (Number of females) / (Total players) = (13 + 33) / 90 = 46/90
* P(J ∩ M') = (Number of female juniors) / (Total players) = 13/90
* P(J ∪ M') = (23/90) + (46/90) - (13/90)
* P(J ∪ M') = 56/90
* P(J ∪ M') ≈ 0.6222
**(d) Probability that a player selected at random is right-handed:**
* P(L') = 1 - P(L)
* P(L') = 1 - (18/90)
* P(L') = 72/90
* P(L') = 0.8
**(e) Probability that a right-handed player selected at random is not a junior:**
* P(S | L') = (Number of right-handed seniors) / (Total number of right-handed players)
* P(S | L') = (34 + 33) / 72
* P(S | L') = 67/72
* P(S | L') ≈ 0.9306
**(f) Probability that a right-handed female player selected at random is a junior:**
* P(J | L' ∩ M') = (Number of right-handed female juniors) / (Total number of right-handed female players)
* P(J | L' ∩ M') = 13 / 33
* P(J | L' ∩ M') ≈ 0.3939
I hope this comprehensive solution is helpful! Let me know if you have any other questions.