Two events e1 and e2 are called independent if p(e1 â© e2) = p(e1)p(e2). for each of the following pairs of events, which are subsets of the set of all possible outcomes when a coin is tossed three times, determine whether or not they are independent. a) e1: tails comes up with the coin is tossed the first time; e2: heads comes up when the coin is tossed the second time. b) e1: the first coin comes up tails; e2: two, and not three, heads come up in a row. c) e1: the second coin comes up tails; e2: two, and not three, heads come up in a row.

 a) Independent

b) Independent

c) Dependent

Step-by-step explanation:

Since, If a coin is tossed three times,

Then, total number of outcomes, n(S) = 8

a)   : tails comes up with the coin is tossed the first time;

= { TTT, THH, THT, TTH }

:  heads comes up when the coin is tossed the second time.

= { THT, HHH, THH, HHT }

Thus,

⇒  

Similarly,  

⇒  

Since,   = { THH, THT }

⇒

Thus,  

Therefore,   and are independent events.

B)   :  the first coin comes up tails

= { TTT, THH, THT, TTH }

:   two, and not three, heads come up in a row

= { HHT, THH }

Thus,

⇒  

Similarly,  

⇒  

Since,   = { THH }

⇒

Thus,  

Therefore,   and are independent events.

C)   :  the second coin comes up tails;

= { HTH, HTT, TTT, TTH }

:   two, and not three, heads come up in a row

= { HHT, THH }

Thus,

⇒  

Similarly,  

⇒  

Since,   =

⇒

Thus,  

Therefore,   and are dependent events.

i think its a and d

step-by-step explanation: