Back to the full dot-point answer

NSWMaths Extension 1Quick questions

Statistical Analysis (ME-S1)

Quick questions on Binomial probability calculations: exact values, cumulative probabilities and complements

15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is exact probability?
Show answer
$P(X=k)=(nk)pk(1−p)n−k,k=0,1,…,n.P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}, \quad k = 0, 1, \dots, n.$
What is cumulative probability (sum of pmf values)?
Show answer
$P(X≤k)=∑i=0k(ni)pi(1−p)n−i.P(X \le k) = \sum_{i = 0}^{k} \binom{n}{i} p^i (1 - p)^{n - i}.$
What is complementary probability?
Show answer
$P(X≥k)=1−P(X≤k−1)=1−∑i=0k−1P(X=i).P(X \ge k) = 1 - P(X \le k - 1) = 1 - \sum_{i = 0}^{k - 1} P(X = i).$
What is standard problem patterns?
Show answer
Exact number of successes: P(X=k)P(X = k) directly from the pmf.
What is choosing the easier sum?
Show answer
If asked P(X≥k)P(X \ge k) and n−k+1n - k + 1 is small, sum directly. If n−k+1n - k + 1 is large, use 1−P(X≤k−1)1 - P(X \le k - 1) (complementary).
What is cumulative?
Show answer
X∼B(4,0.5)X \sim B(4, 0.5). Find P(X≤2)P(X \le 2).
What is complementary?
Show answer
X∼B(10,0.1)X \sim B(10, 0.1). Find P(X≥1)P(X \ge 1).
What is between values?
Show answer
X∼B(6,0.4)X \sim B(6, 0.4). Find P(2≤X≤4)P(2 \le X \le 4).
What is at least one?
Show answer
A coin is flipped 55 times. Find the probability of at least one head.
What is exam item?
Show answer
A class of 2525 students. Each independently has a 20%20\% chance of passing the test. Find the probability that more than 55 students pass.
What is exact number of successes?
Show answer
P(X=k)P(X = k) directly from the pmf.
What is at least one success?
Show answer
P(X≥1)=1−P(X=0)=1−(1−p)nP(X \ge 1) = 1 - P(X = 0) = 1 - (1 - p)^n.
What is no successes?
Show answer
P(X=0)=(1−p)nP(X = 0) = (1 - p)^n.
What is between two values?
Show answer
P(j≤X≤k)=∑i=jkP(X=i)P(j \le X \le k) = \sum_{i = j}^{k} P(X = i).
What is forgetting the binomial coefficient?
Show answer
P(X=k)=(nk)pkqn−kP(X = k) = \binom{n}{k} p^k q^{n - k}, not pkqn−kp^k q^{n - k}.

All Maths Extension 1Q&A pages