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Question 2 (15 marks) Nowadays, it is imperative for us to have a portable te...

Mar 15, 2024

Question 2 (15 marks) Nowadays, it is imperative for us to have a portable telecommunication device such as smart phone, iPad. A survey has been conducted to study the number of portable telecommunication devices owned by a person, and the results are as follows: \begin{tabular}{|l|c|c|c|c|} \hline \begin{tabular}{l} Number of portable telecommunication \\ devices owned (X) \end{tabular} & 1 & 2 & 3 & 4 \\ \hline Number of respondents & 32 & 40 & 12 & 16 \\ \hline \end{tabular} (a) Tabulate the probability distribution of the number of portable telecommunication devices owned by a person (X). (b) Calculate the expected value and variance of the number of portable telecommunication devices owned by a person. (c) For an online shop, the probability of a customer buying an iPad is 0.1 . A random sample of 12 customers are selected. (i) Calculate the probability that at least 3 of them buy an iPad. (ii) Most likely, how many of them buy an iPad?

Solution by Steps

step 1

To tabulate the probability distribution, we first need to calculate the total number of respondents. This is done by summing the number of respondents for each number of devices owned

step 2

The total number of respondents is calculated as:

Total = 32 + 40 + 12 + 16

step 3

The probability of a person owning

X

devices is the number of respondents who own

X

devices divided by the total number of respondents

step 4

The probability distribution table is created by listing the number of devices owned and their corresponding probabilities

Answer

The probability distribution table is as follows: Number of Devices (X) | Probability (P(X)) 1 | 32/100 2 | 40/100 3 | 12/100 4 | 16/100

Key Concept

Probability Distribution

Explanation

The probability distribution shows the probabilities of all possible outcomes for the number of portable telecommunication devices owned by a person.

step 5

To calculate the expected value (mean) of

X

, we multiply each value of

X

by its probability and sum the results

step 6

The expected value is calculated as:

E(X) = (1)(32/100) + (2)(40/100) + (3)(12/100) + (4)(16/100)

step 7

To calculate the variance of

X

, we need to find the expected value of

X^2

and then use the formula

Var(X) = E(X^2) - [E(X)]^2

step 8

First, calculate

E(X^2)

as:

E(X^2) = (1^2)(32/100) + (2^2)(40/100) + (3^2)(12/100) + (4^2)(16/100)

step 9

Then, calculate the variance using the expected values:

Var(X) = E(X^2) - [E(X)]^2

Answer

The expected value (mean) of

X

is 2.2 and the variance of

X

is 1.16.

Key Concept

Expected Value and Variance

Explanation

The expected value is the average number of devices owned, and the variance measures the spread of the distribution around the mean.

step 10

To calculate the probability that at least 3 customers buy an iPad, we use the binomial probability formula

step 11

The binomial probability formula is:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

where

n

is the number of trials,

k

is the number of successes, and

p

is the probability of success

step 12

We need to calculate the probability for

k = 3, 4, ..., 12

and sum these probabilities to find the probability of at least 3 customers buying an iPad

step 13

The probability of at least 3 customers buying an iPad is:

P(X \geq 3) = \sum_{k=3}^{12} \binom{12}{k} (0.1)^k (0.9)^{12-k}

Answer

The probability that at least 3 customers buy an iPad is approximately 0.0134.

Key Concept

Binomial Probability

Explanation

This probability represents the likelihood of a given number of successes in a series of independent and identically distributed Bernoulli trials.

step 14

To find the most likely number of customers to buy an iPad, we look for the value of

k

that maximizes the binomial probability formula for

k = 0, 1, ..., 12

step 15

The most likely number of customers to buy an iPad is the mode of the binomial distribution, which for a binomial distribution with

n

trials and success probability

p

is given by

\lfloor (n+1)p \rfloor

\lceil (n+1)p \rceil - 1

, whichever is a valid value of

k

Answer

The most likely number of customers to buy an iPad is 1.

Key Concept

Mode of a Binomial Distribution

Explanation

The mode is the value that appears most frequently in a set of data, which in the case of a binomial distribution can be found using the given formula.