Probability

Probability That A Randomly Selected Chord Exceeds The Length Of The Radius Of Circle

Situation
If a chord is selected at random on a fixed circle what is the probability that its length exceeds the radius of the circle?

  1. Assume that the distance of the chord from the center of the circle is uniformly distributed.
    A.   0.5 C.   0.866
    B.   0.667 D.   0.75
  2. Assume that the midpoint of the chord is evenly distributed over the circle.
    A.   0.5 C.   0.866
    B.   0.667 D.   0.75
  3. Assume that the end points of the chord are uniformly distributed over the circumference of the circle.
    A.   0.5 C.   0.866
    B.   0.667 D.   0.75

 

1 - Probability for cars to pass through a point on road in a 5-minute period

Problem
The number of cars passing a point on a road may be modelled by Poisson distribution. At an average, 4 cars enters the Caibaan Diversion Road in Tacloban City every 5 minutes. Find the probability that in a 5-minute period (a) two cars go past and (b) fewer than 3 cars go past.
 

Poisson Probability Distribution

The number of occurrences in a given time interval or in a given space can be modeled using Poisson Distribution if the following conditions are being satisfied:

  • The events occur at random.
  • The events are independent from one another.
  • The average rate of occurrences is constant.
  • There are no simultaneous occurrences.

 

The Poisson distribution is defined as

$P(x) = \dfrac{e^{-\mu} \mu^x}{x!}$

where x is a discrete random variable

P(x) = probability for x occurrences
μ = the mean number of occurrences

Two Gamblers Play Until One is Bankrupt: Chance That the Better Player Wins

Problem
Player M has Php1, and Player N has Php2. Each play gives one the players Php1 from the other. Player M is enough better than player N that he wins 2/3 of the plays. They play until one is bankrupt. What is the chance that Player M wins?

A.   3/4 C.   4/7
B.   5/7 D.   2/3

 

Probability of Winning the Carnival Game of Tossing a Coin Into a Table

Problem
In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

A.   5/16 C.   9/256
B.   1/16 D.   3/128

Random Steps of a Drunk Man: Probability of Escaping the Cliff

Problem
From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?

A.   2/27 C.   4/27
B.   107/243 D.   1/2

 

Samuel Pepys Wrote Isaac Newton Asking Which Event is More Likely to Occur

Problem
Samuel Pepys wrote Isaac Newton to ask which of three events is more likely: that a person get (a) at least 1 six when 6 dice are rolled (b) at least two sixes when 12 dice are rolled, or (c) at least 3 sixes when 18 dice are rolled. What is the answer?

A.   (a) is more likely than (b) and (c)
B.   (b) is more likely than (a) and (c)
C.   (c) is more likely than (a) and (b)
D.   (a), (b), and (c) are equally likely

Spinning Spherical Target: Probability for Three Marksmen to Hit on the Same Hemisphere

Problem
Three marksman simultaneously shoot and hit a rapidly spinning spherical target. What is the probability that the three points of impact lie on the same hemisphere?

A.   0 C.   1
B.   1/2 D.   2/3

 

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