Probability and Stats
You flip a fair coin three times. What is the probability of getting exactly two heads?
3/8
You flip a fair coin three times. What is the probability of getting heads after you got 2 heads before?
1/2
If a stock goes from £50 to £75, what’s the percentage increase?
50%
How many zeros are at the end of "100!"?
100/5+100/25 = 24
A square has area 49. What is its perimeter?
Side = 7, Perimeter =28
You roll a fair six-sided die twice. What’s the probability the sum is 7?
6/36 = 1/6
Your investment returns +50% one year and –50% the next. What’s your average annual return and total change?
0% avarage annual return, -25% total change
Simplify 12.5% of 48.
(6)
How many 7-digit numbers (digits 0–9) have no repeated digits and do not start with 0?
9*9*8*7*6*5*4 = 544 320
A metal rod has length L. Its density varies linearly from p0 at one end to p1 at the other. How can you compute its total mass M?
Hints:
1. M=average density * Length
Average density is: (p0+p1)/2,
So Mass=L*(p0+p1)/2
You roll one die. What’s the expected value of the roll?
(1+2+…+6)/6=3.5
Buses come every 10 minutes, but you arrive randomly. On average, how long do you wait?
5 minutes – uniform arrival assumption
48*251=?
48*1000/4 +48 = 12 048
You have 9 identical-looking coins; one is heavier. You have a balance scale and can use it twice. How do you find the heavier coin?
Divide into 3 groups of 3. Weigh two groups. Heavier group contains the coin; weigh two of those coins to find it.
An ice cube melts, and its volume decreases over time as V(t). The melt rate is proportional to its surface area, S(t). Can you set up a dependance equation for V(t) by time (I.e how V(t) depends on t)?
V=s3, S = 6s2, if dV/dt=-kS=-6ks2, substitute s= V1/3 ->dV/dt=-6kV2/3 (or any other non-math notation)
What’s the minimum number of people in a room for the probability that at least two share a birthday to exceed 50%?
23 –> 365/365*364/365*…*(365-n+1)/365, (find n with prob of noone having a birthday)
100 passengers board a plane with assigned seats. The first passenger sits randomly. Each subsequent passenger: Sits in their own seat if available, otherwise in a random empty one. What’s the probability the last passenger sits in their own seat?
1/2 - It doesn’t depend on n!
Reason: only the first and last passenger matter — the last seat left is equally likely to be either the first passenger’s or their own.
Roughly how many cubic meters of air does a person breathe per day? Tidal volume ≈ 0.5 L/breath, ~12 breaths/min.
6*60*24/1000 = 8.6
Four people must cross a bridge at night with one torch; at most two can cross, and they move at the slower person’s speed. What’s the minimum total time? Times are 2, 4, 7, 11.
2+4 cross (4), 2 back (2), 7+11 cross (11), 4 back (4), 2+4 cross (4) → 25 minutes.
You drive 60 km at 60 km/h. How fast must you go on the return trip to average 120 km/h overall?
Impossible (or Infinity)
To average 120 km/h over 120 km total, total time = 1 hour. But the first half already took 1 hour, so you’d need infinite speed back.
Maths notation: 120=(60+60)/1+(60/v2)
Suppose you roll three fair 100-sided die. What is the expected value of the lowest roll?
The probability that X is at least k is (100-k+1)3/1003 so,
E[x]=1003/1003+993/1003+...+13/1003=
=(1+2+...+100)2/1003
rewrite as 50502/1003=25.5025
You have two candles that each burn completely in 1 hour, but they burn non-uniformly (so half length ≠ half time). How can you measure 45 minutes using only these two candles and a lighter?
Light both ends of the first candle and one end of the second at the same time. The first candle burns out in 30 min(because both ends). Then immediately light the other end of the second candle → it now burns out in 15 min.→ Total = 45 minutes.
If an investment grows by 10% each year, approximately how long will it take to double?
rule of 72: 72/10=7 years
Pick two random points on a circle and draw the chord. Is it more likely to be longer than the radius or shorter?
Longer
Most chords exceed the radius
A metal rod has length L. Its density varies from p0 at one end to p1 at the other. How can you compute its total mass M if the density isn’t linear?
Hints:
1. M=average density * Length
2. Consider average of sum p0,p0.001...p0.999,p1
Mass=(Sum p0,p0.001...p0.999,p1 /1000)*L =Int0L p(x)dx