Suppose a telemarketer ha 10% chance of making a sale on any give phon

1.	Suppose a telemarketer ha 10% chance of making a sale on any give phone call. He is required to make 10 successful sales before leaving for day. What is the probability that he needs to make more than 50 calls?
Answer» Probability of success is P = 10% = 10/100 = 1/10 Probability of failure is q = 1 - p = 1 - 1/10 = 9/10 Probability that he required less than or equal to 50 calls to make 10 successful sales is P(E \(\leq\) 50) = P(10) + P(11) + P(12) +.... P(50) = ¹⁰C₁₀(1/10)¹⁰ + ¹¹C₁₀(1/10)¹⁰ 9/10 + ¹²C₁₀(1/10)¹⁰(9/10)²+....+⁵⁰C₁₀(1/10)¹⁰(9/10)⁴⁰ (\(\because\) Minimum 10 calls are required) \(\therefore\) probability that he needs to make more than 50 calls in P(E > 50) = 1 - p(E \(\leq\) 50) = 1 - ((1/10)¹⁰ + ¹¹C₁₀(1/10)¹⁰ 9/10 + ¹²C₁₀(1/10)¹⁰(9/10)²)+ ...+ ⁵⁰C₁₀(1/10)¹⁰(9/10)⁴⁰ = 1 - (1/10)¹⁰(1 + ¹¹C₁₀9/10 + ¹²C₁₀(9/10)²+....+⁵⁰C₁₀(9/10)⁴⁰)

Suppose a telemarketer ha 10% chance of making a sale on any give phone call. He is required to make 10 successful sales before leaving for day. What is the probability that he needs to make more than 50 calls?

Answer»

Probability of success is P = 10% = 10/100 = 1/10

Probability of failure is q = 1 - p = 1 - 1/10 = 9/10

Probability that he required less than or equal to 50 calls to make 10 successful sales

is P(E \(\leq\) 50) = P(10) + P(11) + P(12) +.... P(50)

= ¹⁰C₁₀(1/10)¹⁰ + ¹¹C₁₀(1/10)¹⁰ 9/10 + ¹²C₁₀(1/10)¹⁰(9/10)²+....+⁵⁰C₁₀(1/10)¹⁰(9/10)⁴⁰

(\(\because\) Minimum 10 calls are required)

\(\therefore\) probability that he needs to make more than 50 calls in P(E > 50) = 1 - p(E \(\leq\) 50)

= 1 - ((1/10)¹⁰ + ¹¹C₁₀(1/10)¹⁰ 9/10 + ¹²C₁₀(1/10)¹⁰(9/10)²)+ ...+ ⁵⁰C₁₀(1/10)¹⁰(9/10)⁴⁰

= 1 - (1/10)¹⁰(1 + ¹¹C₁₀9/10 + ¹²C₁₀(9/10)²+....+⁵⁰C₁₀(9/10)⁴⁰)