4.8 Binomial distributionIB Maths: Applications and Interpretation HL: Revision notes

A binomial distribution models the number of successes in a fixed number of trials. It is appropriate when: there is a fixed number $n$ of trials; each trial has only two outcomes, success and failure; the probability of success $p$ is the same each time; and the trials are independent. Examples: the number of correct guesses on a multiple-choice quiz, free throws scored, or defective bulbs in a box. We write $X\sim\mathrm{B}(n,p)$.

State the distribution clearly, with $n$ and $p$ from the context. A quiz of 12 questions guessed at random with four options each gives $X\sim\mathrm{B}(12,0.25)$. 'Success' is whatever you are counting, even if it is bad news, such as a defective bulb with $p=0.04$. The probability of exactly $x$ successes is $P(X=x)=\binom{n}{x}p^x(1-p)^{n-x},$ but in examinations you should find binomial probabilities using technology.

Use binomial pdf for $P(X=x)$ and binomial cdf for $P(X\leq x)$. Enter $n$, $p$ and $x$. For $X\sim\mathrm{B}(12,0.25)$, $P(X=3)=0.258$. Convert other inequalities into 'less than or equal':

• $P(X\geq k)=1-P(X\leq k-1)$
• $P(a\leq X\leq b)=P(X\leq b)-P(X\leq a-1)$
• $P(X<k)=P(X\leq k-1)$ Example: $S\sim\mathrm{B}(15,0.35)$, $P(S\geq8)=1-P(S\leq7)=0.113$. To find the smallest $k$ with $P(X\leq k)>0.9$, use a table of cumulative values and look for where it first passes $0.9$.

For $X\sim\mathrm{B}(n,p)$: $\mathrm{E}(X)=np,\qquad\mathrm{Var}(X)=np(1-p).$ For $\mathrm{B}(8,0.7)$, the mean is $5.6$ and the variance is $1.68$. The mean is the expected number of occurrences from SL 4.5: with $n$ trials each with probability $p$, you expect $np$ successes. A formal proof of these results is not required.

A factory's bulbs are defective with probability $0.04$, in boxes of 25. Let $D\sim\mathrm{B}(25,0.04)$. Then $P(D=0)=0.96^{25}=0.360$ and $P(D\geq2)=1-P(D\leq1)=0.264$. Interpret in context: about 26% of boxes contain two or more defective bulbs. In a question, follow the sequence: define the variable, state $\mathrm{B}(n,p)$, rewrite the probability in terms of $P(X=x)$ or $P(X\leq x)$, use the GDC, and give the answer to 3 s.f. Where the question asks about the model, name two conditions: fixed number of independent trials, constant probability of success.