Question

A string quartet consists of two violinists, a violist, and a cellist. A survey of 100 string quartets (400 musicians) reported that 98 cellists had no violin experience and 74 violists had violin experience.
1. What is the probability of selecting a musician who plays or had played the violin?
2. Conditional on selecting a non-cellist, what is the probability of selecting someone who neither plays nor played the violin?
3. Suppose you want to study the monetary implications of switching instruments. What is the minimum probability of selecting a quartet in which at least one player has switched? State, make, and use minimal assumptions as needed.

Answer 1

Out of 400 musicians
Violinist = 200
Violist = 100
Cellists = 100

	Violin Experience	No Violin Experience	Total
Violist	74	16	100
Cellist	2	98	100

1) Number of musicians who have play or have played violin       
   = 200 + 74 + 2 = 276     
P(selecting a musician who plays or has played violin)      
   = 276 / 400     
   = 0.69     

P(selecting a musician who plays or has played violin)   = 0.69

2) Number of non-cellists = 400 - 100 = 300      
Number of non-cellists who neither play nor played the violin = 16      
P(selecting someone who neither plays nor played the violin | he is a non-cellist)       
   = 16/300     
   = 0.0533     
P(selecting someone who neither plays nor played the violin | he is a non-cellist) = 0.0533

3) For selecting a quarted, we have to select 2 violinist, 1 violist and 1 cellist.      
Assuming a player can switch to being a violinist if he has violin experience,      
we have (74+2 = 76) people who can switch to being a violinist.      
Also, assumption is that no person other than the player stated in the above assumption      
can switch      
P(selecting a quartet in which at least one player has switched)      
= 1 - P(selecting a quartet in which no player has switched)      
4 persons can be selected from 400 in 400C4 = 1050739900 ways      
2 violinists (who have not switched) can be selected in 200C2 = 19900      
1 cellist who cannot switch can be selected in 98C1 = 98 ways      
1 violist who cannot switch can be selected in 16C1 = 16 ways      
       
P(selecting a quartet in which no player has switched) =       
       
             = 0.0297  
P(selecting a quartet in which at least one player has switched) = 1 - 0.0297      
       = 0.9703
P(selecting a quartet in which at least one player has switched) = 0.9703