Newsfeed Ads - Binomial & Poisson Approx.
Each of the 100 stories in a newsfeed has an independent 4% chance of showing an advertisement. What is the probability that exactly 1 ad appears in the entire newsfeed?
(Show both the exact Binomial calculation and the Poisson approximation.)
Related Concepts
Hint
We need to calculate the probability of exactly 1 ad (k=1) using two methods.
- For the Exact Binomial Calculation:
- Number of trials (stories), n = 100.
- Probability of success (ad on a story), p = 4% = 0.04.
- Probability of failure (no ad), 1-p = 1 - 0.04 = 0.96.
- We want P(X = 1). Use the Binomial formula:
P(X = 1) = C(n, 1) * p1 * (1-p)n-1 - Remember that C(n, 1) = n. So, C(100, 1) = 100.
- You'll need to calculate (0.96)99.
- For the Poisson Approximation:
- First, check if the approximation is reasonable: Is 'n' large? (n=100 is pretty large). Is 'p' small? (p=0.04 is small). Is np = λ not too large? (np = 100 * 0.04 = 4, which is fine). So, yes, Poisson approximation should work well.
- Calculate the average rate for the Poisson distribution: λ = n * p.
- We want P(X = 1). Use the Poisson formula:
P(X = 1) ≈ (e-λ * λ1) / 1! - Remember 1! = 1.
- You'll need to calculate e-λ.
- Calculator Use: You will need a calculator for (0.96)99 and for e-λ. For example, e-4 is approximately 0.0183156.
Explanation: Newsfeed Ads - Binomial & Poisson Approximation
Imagine you're scrolling through a newsfeed with 100 different stories. For each story, there's a small 4% chance that an advertisement will pop up. We want to figure out the probability that you see exactly one advertisement in total while scrolling through all 100 stories.
This is a classic "Binomial" problem because:
- There's a fixed number of tries (100 stories).
- Each story (try) has two outcomes: either an ad appears (a "success") or it doesn't (a "failure").
- The chance of an ad is the same for every story (4% or 0.04).
- Whether an ad appears on one story doesn't affect whether it appears on another (they are independent).
We will calculate this probability in two ways:
- The Exact Way (Binomial Formula): This gives the precise answer.
- An Easier Shortcut (Poisson Approximation): Because we have many tries (100 stories) and the chance of an ad on any single story is small (4%), we can use a simpler formula (Poisson) that gives a very close answer. This is helpful because the Binomial formula can get complicated with large numbers.
Let X be the random variable representing the number of ads that appear in the 100-story newsfeed.
- Number of trials (stories): n = 100.
- Probability of success (an ad appearing on one story): p = 0.04.
- Probability of failure (no ad on one story): 1-p = 1 - 0.04 = 0.96.
So, X follows a Binomial distribution: X ~ Binomial(n=100, p=0.04).
We want to find the probability of exactly 1 ad, so k = 1.
1. Exact Calculation using Binomial Distribution (P(X = 1))
The Binomial probability formula is: P(X = k) = C(n, k) * pk * (1-p)n-k
For k=1:
Step 1: Calculate C(n, k) = C(100, 1) C(100, 1) = 100! / (1! * (100-1)!) = 100! / (1! * 99!) This simplifies to C(100, 1) = 100. (There are 100 ways to choose which single story shows the ad).
Step 2: Calculate pk p1 = (0.04)1 = 0.04.
Step 3: Calculate (1-p)n-k (1-p)n-k = (0.96)100-1 = (0.96)99. Using a calculator: (0.96)99 ≈ 0.0183070 (keeping several decimal places for accuracy).
Step 4: Multiply all parts together P(X = 1) = C(100, 1) × (0.04)1 × (0.96)99 P(X = 1) ≈ 100 × 0.04 × 0.0183070 P(X = 1) ≈ 4 × 0.0183070 P(X = 1) ≈ 0.073228
So, the exact probability using the Binomial distribution is approximately 0.07323.
2. Using the Poisson Approximation
Since 'n' is large (100) and 'p' is small (0.04), we can use the Poisson distribution as an approximation. First, calculate the average number of ads expected, λ (lambda): λ = n × p λ = 100 × 0.04 = 4
So, we are approximating with a Poisson distribution with λ = 4. The Poisson probability formula is: P(X = k) ≈ (e-λ * λk) / k!
We want to find P(X = 1), so k = 1:
Step 1: Calculate e-λ e-λ = e-4. Using a calculator: e-4 ≈ 0.0183156.
Step 2: Calculate λk λk = 41 = 4.
Step 3: Calculate k! k! = 1! = 1.
Step 4: Put it all together P(X = 1) ≈ (e-4 * 41) / 1! P(X = 1) ≈ (0.0183156 * 4) / 1 P(X = 1) ≈ 0.0732624
So, using the Poisson approximation, the probability is approximately 0.07326.
Comparison and Final Answer
Let's compare the results:
- Exact Binomial calculation: P(X = 1) ≈ 0.073228
- Poisson Approximation: P(X = 1) ≈ 0.073262
Both values are very close! Rounding to three significant figures (or three decimal places as often requested):
P(Exactly 1 ad) ≈ 0.073
This means there is approximately a 7.3% chance that exactly one ad will appear in the entire newsfeed.
Why does the Poisson approximation work well here? The conditions are good: 'n' (100) is large, 'p' (0.04) is small, and their product λ = np = 4 is a moderate number. The Poisson distribution is often used for "rare events" happening over many opportunities, which is what we have here (an ad is a somewhat rare event for each story, but there are many stories).
The approximation makes calculations easier, especially if you didn't have a calculator that could easily handle (0.96)99 or large combinations directly.
What if? Using the Poisson approximation (λ=4), what is the probability of seeing no ads (P(X=0)) in the newsfeed? How would you calculate P(X=0) using the exact Binomial formula? Compare the results!