Binomial Probability Distribution: A Case Study of HotelsByUs Scenario
An interactive survey for HotelsByUs asked respondents, “When travelling internationally, do you generally venture out on your own to experience the local culture, or stick with your tour group and itineraries?” The survey found that 23% of the respondents stick with their tour group.
Why Binomial Distribution applies here:
- Binary outcomes: There are two possible outcomes for each traveler (success or failure).
- Fixed number of trials: There is a fixed number of trials (6 travelers).
- Constant probability of success: The probability of success is constant (23%) for each traveler.
- Independence of trials: The trials are independent (the outcome for one traveler doesn’t affect others).
The binomial probability formula is expressed as:
Alternatively, if you prefer the expanded form of the binomial coefficient (especially in the absence of a statistical calculator), you can use the following expression:
Where:
- ( P(X = x) ): The probability of exactly ( x ) successes.
- ( n ): Total number of trials
- ( x ): Number of successes.
- ( p ): Probability of success on a single trial.
- ( 1-p ): Probability of failure on a single trial.
Step-by-Step Calculation:
Step 1: Define the variables:
$n = 6$ (total number of respondents).
$p = 0.23$ (the probability that the respondents stick with their tour group ).
$x = 2$ (number of successes)
Step 2: Now, apply the binomial probability formula:
$$P(X = 2) = {_{6}C_2} \cdot 0.23^2 \cdot (1-0.23)^{6-2}$$
$$
P(X = 2) = \frac{6!}{6!(6 \;- \; 2)!} \times 0.23^{2} \times (0.77)^{4} = \frac{6 \times 5}{2 \times 1} \times (0.23)^2 \times (0.77)^4 = 15 \times 0.0529000 \times 0.3515304 = 0.2789394
$$
The probability that exactly two out of the six travelers will stick with their tour group is approximately (0.2789) or (27.89\%).
Step-by-Step Calculation:
Step 1: Define the variables:
$n = 6$ (total number of respondents).
$p = 23\% = 0.23$ (the probability that the respondents stick with their tour group ).
We are looking for the probability that at least two travellers will stick with their tour group. This is expressed as $P(X \geq 2)$, where $X$ is the number of successes (travelers who stick with the tour group)
Step 2: Probability of At Least Two
To calculate $P(X \geq 2)$, we calculate the individual probabilities for $P(X = 0), \; P(X = 1)$ and then subtract them from 1.
$$P(X \geq 2) = 1 – P(X < 2)$$
$$P(X \geq 2) = 1 – [P(X = 0) + P(X = 1)]$$
$$
P(X = 0) = {_{6}C_0} \cdot 0.23^0 \cdot (1-0.23)^{6-0}
$$
$$
P(X = 0) = \frac{6!}{0!(6 \;- \; 0)!} \times 0.23^{0} \times (0.77)^{6} = \frac{1}{1 \times 1} \times 1 \times (0.77)^6 = 1 \times 1 \times 0.20842238 = 0.20842238
$$
$$
P(X = 1) = {_{6}C_1} \cdot 0.23^1 \cdot (1-0.23)^{6-1}
$$
$$
P(X = 1) = \frac{6!}{1!(6 \;- \; 1)!} \times 0.23^{1} \times (0.77)^{5} = \frac{6}{1 \times 1} \times 0.23 \times (0.77)^5 = 6 \times 0.23 \times 0.27067842 = 0.37353621
$$
Step 3: Calculate Probability of At Least Two
$$P(X \geq 2) = 1 – [0.20842238 + 0.37353621] = 0.41804141$$
The probability that at least two travelers will stick with their tour group is approximately $0.41804141$ or $41.80\%$.
The calculated probability of $0.4180$ or $41.80\%$ means that, based on the given sample size of six international travellers, there is a (41.8\%) chance that at least two travellers will stick with their tour group while the others venture out to experience the local culture. In other words, almost half of the time, you can expect that at least two people in a group of six would follow the tour group instead of exploring independently. This information could be useful for understanding group dynamics, marketing, and planning for travel-related businesses like HotelsByUs.