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Probability distributions, binomial distribution, maths grade 11/12

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Study note

Ahmed Nour ✓™

United States of America

DEFAULT (US)

Statistics

Statistical distributions

Probability, random variables, and probability distributions

The binomial distribution

Probability Distributions
random variable i'S
sumple space
१८
1
2 3 4 56
P(X=x) 1/6 1/6 1/6 1/16 1
x
1
P(SS) 26
This spinner is spun until it lands on red or has been spun four times in total.
Find the probability distribution of the random variable S, the number of
times the spinner is spun.
P/R)= 2/5 P(B) = 3/5
2 3 4
18 27
5 25 125 125
a)
a P(X= 1)
b P(X= 28)
e P(13 < X<42)
I eg. a dice being volled
a) P(X=1) = 1/50 ✓
b) P(x ≥28) = 23/50
a P(1 < X <3)
b P(X<2)
c P(X> 3)
10 A discrete random variable X has the
probability distribution shown in this table.
Find:
or y
b) P(X<2)
P
9 The random variable X can take any integer value from 1 to 50. Given that X has a discrete
uniform distribution, find:
O
P(x=x) 0.6
x
a) P(1 ≤ x ≤ 3) = P(X=2) + P(X=3)
→ probability that random variable x takes.
value x is P(X= x)
c) P(134X<42)
P(X= x)
1
0.64 x
0.4
= P(X=1) + P(X=0)
1/4 +
+ 1/8 = 3/8
12 A fair five-sided spinner is spun.
Given that the spinner is spun five times, write down, in table form,
the probability distributions of the following random variables:
a X, the number of times red appears
bY, the number of times yellow appears.
½/2 + 1/8 5/8
1st =
2nd = 3/5x2/5
3rd = 3/5x3/5x²/5
2/5
= 27/125
4th=1-18-6-2 as it's end of experiment
125 25
0
3
The spinner is now spun until it lands on blue, or until it has been spun five times.
The random variable Z is defined as the number of spins in this experiment.
e Find the probability distribution of Z.
1
1
2
3
0.63 0.6²x
*0.4²
0.43
B
B
R
R
28150
B
R
2
3
B
Y
(1 mark)
(1 mark)
(1 mark)
c) P(X73) = O✓
B
3
7
=
4
0.6 x
0.4
1425✓
5
0.45 Binomial Distribution.
probability
sucess
B~~ (۸۲)
number of
trials
a)
The random variable X~B(12,). Find:
a P(X= 2)
b P(X=9)
P/x = √) = nCrx prx (1-P)^-*
2
12 C 2 x 1/16
=
Sucess
c) P(X ≤ 1) = P(X=0) + P(X=1)
5¹²
= 0.381
x 5
6
10
+
c) P(x≤2)
- sucess / faliure aka independant.
-fixed no. of trials and probability
foliure
= 0.296
E/P 8 A dice is biased so that the probability of it landing on a six is 0.3. Hannah rolls the dice
15 times.
=
a P(X> 20)
c P(15 X < 20)
c P(X ≤ 1)
a State any assumptions that are required to model the number of sixes as a binomial
distribution. State the distribution.
12C1x¹x 5"
b Find the probability that Hannah rolls exactly 4 sixes.
e Find the probability that she rolls two or fewer sixes.
a) rowed a set amount of times (15)
sucess of faliure (6 or anything else)
variables are independant
b) P(x =4)
= 15C4 x 0.34 x 0.7" = 0.219
4 The random variable X~ B(37, 0.65). Find:
b P(X26
d P(X= 23)
= P(x=0) + P(x=1) + P(x=2)
0.715 +15C1 +0.3¹x0.7¹4 + 15C 2 x0.32x0.713
1- P(x≤7)
Cumulative Binomial
→ finding probability x is less than or equal to something
eg. P(X ≤3)
or P(x28)
a) P(x720) 1- - P(x ≤20)
CALCULATOR
Binomial PD for
P(x = a value)
Binomial CD for P(x ≤ a value)
(2 marks)
(2 marks)
(3 marks)
=
= 1-0-11177 = 0.888
b) P(x≤26) = 0.799
c) P(154x420)
= P(X≤19) - P(X ≤14) = 0.06061
=
0.127
6.846x10= 0.0599 d) P(x=23)
0.1258
9 In a town, 30% of residents listen to the local radio.
Ten residents are chosen at random.
X= the number of these ten residents that listen to the local radio.
a Suggest a suitable distribution for X and comment on any necessary assumptions.
b Find the probability that at least half of these 10 residents listen to local radio.
e Find the smallest value of s so that P(X>s) <0.01.
a) X~B (10,0.3)
→ sucess/ falivre → independant → fixed no. of trials and probability
b) P(x>5)
= 1 - P(X ≤4)
= 10.8497 = 0.150268
c) P(X28)
= 1- P(x ≤7) =
1.59x10-320.01
(2 marks)
(2 marks)
(2 marks)
P(X>6)
1- P(x≤5) = = 0.047 >0.01
P(x7) = 0.0105 >0.01
.. smallest value of S=8

Statistical distributions

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Statistical distributions

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying