Three coins are tossed. What is the probability of getting (i) all heads, (ii) two heads, (iii) at least one head, (iv) at least two heads?


Solution:

Let ‘S’ be the sample – space. Then S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }
(i) Let ‘E1’ = Event of getting all heads, Then E1 = { HHH }
 |E1| = 1
 P(E1) = |E1| / |S| = 1 / 8
(ii) Let E2 = Event of getting ‘2’ heads. Then:
 E2 = { HHT, HTH, THH }
 |E2| = 3
 P (E2) = 3 / 8
(iii) Let E3 = Event of getting at least one head. Then:
 E3 = { HHH, HHT, HTH, THH, HTT, THT, TTH }
 |E3| = 7
 P (E3) = 7 / 8
(iv) Let E4 = Event of getting at least one head, Then:
 E4 = { HHH, HHT, HTH, THH, }
 |E4| = 4
 P (E4) = 4/8 = 1/2

A school contains students in grades 1, 2, 3, 4, 5, and 6. Grades 2, 3, 4, 5, and 6 all contain the same number of students, but there are twice this number in grade 1. If a student is selected at random from a list of all the students in the school, what is the probability that she will be in grade 3?

Operator norm convergence

Prove (show) that operator norm convergence implies strong convergence.