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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
Total number of respondents = 32 + 40 + 12 + 16
step 3
The probability of a person owning X X devices is the number of respondents who own X X devices divided by the total number of respondents
step 4
Probability distribution: P(X=1)=32100 P(X=1) = \frac{32}{100} , P(X=2)=40100 P(X=2) = \frac{40}{100} , P(X=3)=12100 P(X=3) = \frac{12}{100} , P(X=4)=16100 P(X=4) = \frac{16}{100}
Answer
Probability distribution: P(X=1)=0.32 P(X=1) = 0.32 , P(X=2)=0.40 P(X=2) = 0.40 , P(X=3)=0.12 P(X=3) = 0.12 , P(X=4)=0.16 P(X=4) = 0.16
Key Concept
Probability Distribution
Explanation
The probability distribution shows the probabilities of all possible outcomes of a random variable.
step 5
To calculate the expected value (mean) of X X , we multiply each value of X X by its corresponding probability and sum the products
step 6
Expected value \( E(X) = (1)(0.32) + (2)(0.40) + (3)(0.12) + (4)(0.16)
step 7
To calculate the variance of X X , we need to find E(X2) E(X^2) first, which is the sum of each value of X X squared multiplied by its corresponding probability. Then we use the formula Var(X)=E(X2)−[E(X)]2 \text{Var}(X) = E(X^2) - [E(X)]^2
step 8
Calculate E(X2)=(12)(0.32)+(22)(0.40)+(32)(0.12)+(42)(0.16) E(X^2) = (1^2)(0.32) + (2^2)(0.40) + (3^2)(0.12) + (4^2)(0.16) and then the variance
Answer
Expected value E(X)=2.2 E(X) = 2.2 , Variance Var(X)=1.36 \text{Var}(X) = 1.36
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 9
To calculate the probability that at least 3 customers buy an iPad, we use the binomial distribution formula P(X≥k)=1−∑i=0k−1P(X=i) P(X \geq k) = 1 - \sum_{i=0}^{k-1} P(X = i) , where P(X=i) P(X = i) is the probability of exactly i i successes in n n trials
step 10
The probability of exactly i i customers buying an iPad is given by P(X=i)=(ni)pi(1−p)n−i P(X = i) = \binom{n}{i} p^i (1-p)^{n-i} , where n=12 n = 12 , p=0.1 p = 0.1 , and i i ranges from 0 to 2 for our calculation
step 11
Calculate P(X=0) P(X = 0) , P(X=1) P(X = 1) , and P(X=2) P(X = 2) using the binomial formula
step 12
Use the results to find P(X≥3)=1−(P(X=0)+P(X=1)+P(X=2)) P(X \geq 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
Answer
Probability that at least 3 customers buy an iPad is approximately 0.0134
Key Concept
Binomial Distribution
Explanation
The binomial distribution is used to model the number of successes in a fixed number of independent trials with the same probability of success.
step 13
To find the most likely number of customers to buy an iPad, we look for the value of i i that maximizes the binomial probability P(X=i) P(X = i) . This is the mode of the distribution
step 14
For a binomial distribution with n=12 n = 12 and p=0.1 p = 0.1 , the mode is typically ⌊(n+1)p⌋ \lfloor (n+1)p \rfloor or ⌈(n+1)p⌉−1 \lceil (n+1)p \rceil - 1 , whichever is a valid number of successes
Answer
Most likely, 1 customer buys an iPad.
Key Concept
Mode of Binomial Distribution
Explanation
The mode is the most frequent outcome in a probability distribution, which in the case of a binomial distribution can be approximated as described.
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