IB Math AI SL/HL Paper 1 Exam Style Practice Questions: Probability-Binomial Distribution
Binomial Distribution — Blood Type O
IB Mathematics HL • Probability • Binomial Distribution • Expected Value and Standard Deviation
Lesson Overview
In this lesson, we model the number of babies with blood type O using a binomial distribution. This is a standard IB probability context because each baby either has blood type O or does not, and the probability remains constant from birth to birth.
We are told that the probability a baby has blood type O is
![\[p=0.46\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-db9e497e69ae4cc58edf9c2ea9928fbc_l3.png?resize=74%2C18&ssl=1 "Rendered by QuickLaTeX.com")
and the hospital considers the next
![\[8\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-02dc48eea604628cf071410ff1d131a6_l3.png?resize=10%2C14&ssl=1 "Rendered by QuickLaTeX.com")
babies born.
If 
is the number of these 8 babies who have blood type O, then
![\[X\sim B(8,0.46)\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-7ca08604925212dbb263076620053483_l3.png?resize=130%2C20&ssl=1 "Rendered by QuickLaTeX.com")
⭐ Key Concepts
- A binomial distribution applies when there is:
- a fixed number of trials,
- only two possible outcomes on each trial,
- a constant probability of success,
- independent trials.
- The binomial probability formula is
![\[P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-821fc3da83820e30dda8d6826b75b0de_l3.png?resize=261%2C48&ssl=1 "Rendered by QuickLaTeX.com")
- The expected value of a binomial distribution is
![\[E(X)=np\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-e7f75833992c48b55997bbee5cbba0b5_l3.png?resize=99%2C20&ssl=1 "Rendered by QuickLaTeX.com")
- The standard deviation is
![\[\sigma=\sqrt{np(1-p)}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-398df9f45978ed630acd55ae0bb70a05_l3.png?resize=142%2C24&ssl=1 "Rendered by QuickLaTeX.com")
📘 Clear IB-Style Explanation
In this question, a “success” means that a baby has blood type O.
Since each baby either has blood type O or does not, there are only two outcomes per trial. The number of babies being considered is fixed at 8, and the probability of success is constant at 0.46.
Therefore, a binomial model is appropriate:
IB questions on the binomial distribution usually ask for:
- individual probabilities such as “exactly 5”,
- cumulative probabilities such as “at least 6”,
- the expected value,
- the standard deviation.
📌 Worked Example 1 — Conditions for a Binomial Distribution
A hospital records whether babies born during a certain month have blood type O. The probability that a baby has blood type O is 0.46, and blood types are assumed to be independent.
(a) State one criterion, in addition to independence, that supports using a binomial distribution.
Solution
One valid criterion is that the probability of success remains constant for each trial.
So, one acceptable answer is:
The probability that each baby has blood type O is constant, with 
for every birth.
Other acceptable criteria include:
- there is a fixed number of trials, namely 8 babies,
- each trial has only two outcomes: blood type O or not blood type O.
📊 Marks: A1
📌 Worked Example 2 — Binomial Probabilities
Let
(a) Find the probability that exactly 5 of the 8 babies have blood type O.
(b) Find the probability that at least 6 of the 8 babies have blood type O.
Solution
(a) Probability that exactly 5 babies have blood type O
![\[P(X=5)=\binom{8}{5}(0.46)^5(0.54)^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-309819401fe2c49024bcaa7eb64b3f91_l3.png?resize=271%2C48&ssl=1 "Rendered by QuickLaTeX.com")
![\[=56(0.46)^5(0.54)^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-1a0b7f9056db9ed76485faf26a9c7260_l3.png?resize=160%2C23&ssl=1 "Rendered by QuickLaTeX.com")
![\[\approx0.1314\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-d66f49c8d9eaf0c31248f1569b59ea5c_l3.png?resize=76%2C14&ssl=1 "Rendered by QuickLaTeX.com")
So
![\[P(X=5)\approx0.131\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-fe621bbc10a539897f91af1c3714f316_l3.png?resize=158%2C20&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 A1
(b) Probability that at least 6 babies have blood type O
“At least 6” means
![\[P(X\ge6)=P(X=6)+P(X=7)+P(X=8)\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-af1c8bf40c03b302d5e7aeab13d13e5e_l3.png?resize=417%2C20&ssl=1 "Rendered by QuickLaTeX.com")
Now calculate each term or use TI84 or any GDC:
![\[P(X=6)=\binom{8}{6}(0.46)^6(0.54)^2\approx0.0419\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-aa45cf2067a448818975825ee3cf653f_l3.png?resize=355%2C48&ssl=1 "Rendered by QuickLaTeX.com")
![\[P(X=7)=\binom{8}{7}(0.46)^7(0.54)\approx0.0102\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-5fa000dcaef5b7334923cbb504f5136a_l3.png?resize=344%2C48&ssl=1 "Rendered by QuickLaTeX.com")
![\[P(X=8)=\binom{8}{8}(0.46)^8\approx0.00199\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-06b10f1e2b4038022c229baeabb9e94c_l3.png?resize=305%2C48&ssl=1 "Rendered by QuickLaTeX.com")
Hence
![\[P(X\ge6)\approx0.0419+0.0102+0.00199\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-b7c2f4b09a4fbc60d5cde2eee78a2928_l3.png?resize=339%2C20&ssl=1 "Rendered by QuickLaTeX.com")
![\[\approx0.0541\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-3e9016bdc341bc660fecb2632184face_l3.png?resize=75%2C14&ssl=1 "Rendered by QuickLaTeX.com")
So
![\[P(X\ge6)\approx0.054\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-4bae90152e4d9f77005b0db0d5931a58_l3.png?resize=159%2C20&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 M1 A1
📌 Worked Example 3 — Expected Value and Standard Deviation
Let
(a) Find the expected number of babies, out of 8, who have blood type O.
(b) Find the standard deviation.
Solution
(a) Expected value
![\[=8(0.46)=3.68\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-76647b60932fcf232d7f8ae138cd51f4_l3.png?resize=144%2C20&ssl=1 "Rendered by QuickLaTeX.com")
So the expected number is
![\[3.68 = 4\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-58075d220f3d01b811128040450a6635_l3.png?resize=73%2C14&ssl=1 "Rendered by QuickLaTeX.com")
babies
📊 Marks: M1 A1
(b) Standard deviation
![\[=\sqrt{8(0.46)(0.54)}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-d64c5a511b50af45c9d48b53c13a340f_l3.png?resize=153%2C24&ssl=1 "Rendered by QuickLaTeX.com")
![\[=\sqrt{1.9872}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-ba2331bb3e5596b7570e932458310ed5_l3.png?resize=93%2C20&ssl=1 "Rendered by QuickLaTeX.com")
![\[\approx1.41\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-601d7da34121e4c623c7b4a565862571_l3.png?resize=55%2C14&ssl=1 "Rendered by QuickLaTeX.com")
So the standard deviation is
![\[1.41\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-d3b32dc9b362c0fc4d99433e4f7710ce_l3.png?resize=34%2C14&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 A1
⚠ Common Mistakes
- Using the wrong value for
instead of writing 
. - Forgetting that “at least 6” means adding
, 
, and 
. - Mixing up the formulas for expected value and standard deviation.
- Using
instead of 
for the standard deviation. - Giving a criterion for binomial distribution that does not actually apply to the situation.
📘 IB Exam Tips
- Always define the random variable clearly, for example “Let be the number of babies with blood type O”.
- Write the
