The sides of an isosceles triangle are whole numbers, and its perimeter is 30 units. What is the probability that the triangle is equilateral?

Brenton Dixon 2022-07-21 Answered
Probability That the Sides of an Isosceles Triangle is an Equilateral Triangle
The sides of an isosceles triangle are whole numbers, and its perimeter is 30 units. What is the probability that the triangle is equilateral?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Kali Galloway
Answered 2022-07-22 Author has 16 answers
Step 1
There aren't a lot of options. There is only one equilateral triangle, which has side length (10,10,10).
In order to form a triangle, the sum of any two sides needs to be larger than the third side, so only these combinations are possible: (14,14,2),(13,13,4),(12,12,6),(11,11,8),(10,10,10),(9,9,12),(8,8,14)
Step 2
So the answer is 1 7
Not exactly what you’re looking for?
Ask My Question
Deromediqm
Answered 2022-07-23 Author has 2 answers
Step 1
Let a,b,c denote the lengths of the sides of our triangle. Without loss of generality, since the triangle is isoceles, let a = c, so that (a,a,b) describes our triangle, where a , b N . Now we know additionally 2 a + b = 30, and by the triangle inequality, b 2 a, So 30 2 a 2 a, i.e. a 8, but of course a 15. Thus the possible triangles are described by { ( 8 , 8 , 14 ) , ( 9 , 9 , 12 ) , ( 10 , 10 , 10 ) , ( 11 , 11 , 8 ) , ( 12 , 12 , 6 ) , ( 13 , 13 , 4 ) , ( 14 , 14 , 2 ) , ( 15 , 15 , 0 ) }.
Step 2
Only one of these is equilateral. Now assuming uniform distribution over these possibilities, and depending whether you include the degenerate (15,15,0), the probability is either 1 7 or 1 8 .
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-18
Geometric distribution where failure probability is not 1 p
The typical geometric distribution is defined from the success probability p, i.e., a r.v. G~Geometric(p), would have PMF... P [ G = g ] = ( 1 p ) g 1 p
Fischer and Spassky play a chess match in which the first played to win a game wins the match. After 10 successive draws. the match is declared drawn. Each game is won by Fischer with probability 0.4 is won by Spassky with probability 0.3. and is a draw with probability 0.3. independent of previous games.
a) What is the probability that Fischer wins the match?
For part a, the answer is obviously Σ k = 1 10 ( 0.3 ) k 1 ( 0.4 ) 0.571.
But we could also say that r.v. L~CustomGeometric( p = 0.4, q = 0.3)...
P [ L = l ] = q l 1 p = ( 0.3 ) l 1 ( 0.4 )
Where the answer to part a would just come from evaluating the CDF of L, i.e., P [ L 10 ]. Is there a name for this more generic version of the Geometric distribution?
asked 2022-08-19
Branching Process - Extinction probability geometric
Consider a branching process with offspring distribution Geometric( α); that is, p k = α ( 1 α ) k for k 0.
a) For what values of α ( 0 , 1 ) is the extinction probability q = 1.
Let { Z n } n 0 be a branching process with p 0 > 0. Let μ = k 0 k p k be the mean of the offspring distribution and let g ( s ) = k 0 s k p k be the probability generating function of the offspring distribution.
- If μ 1, then the extinction probability q = 1.
- If μ > 1, then the extinction probability q is the unique solution to the equation s = g ( s ) with s ( 0 , 1 ).
b) Use the following proposition to give a formula for the extinction probability of the branching process for any value of the parameter α ( 0 , 1 ).
asked 2022-07-23
Probability with elements of geometry
We have a wire of length 20. We bend this wire in random point. And then bend again to get rectangual frame. What's the probability that area of this rectangle is less than 21.
asked 2022-08-18
Probability function of Y = m a x { X , m } for m positive integer when X is geometric distribution
X has geometric distribution so f X ( x ) = p ( 1 p ) x . I wrote this:
f Y ( y ) = P ( Y = y ) = P ( m a x { X , m } = y )
So if y = m it means that X < m so f Y ( y ) = P ( X < m ) = 1 P ( X m ) = 1 ( 1 p ) m and if y = m + 1 , m + 2 , . . . then f Y ( y ) = p ( 1 p ) y
Is it right?
asked 2022-08-26
At a given distance from the origin, which convex subsets of an p -ball have the maximal volume?
For a positive integer n and p [ 1 , ], let B n , p := { x R n x p 1 } be the p unit-ball in R n . Fix h [ 0 , 1 ].
Of all convex subsets A of B n , p with d ( 0 , A ) = h, which shape maximizes the volume ?
asked 2022-07-13
Probability of weather on consecutive days.
Probability of a cloudy day is .55 Probability of a sunny day is .45
A)What is the probability of three consecutive cloudy days, followed by a sunny day?
B)What is the probability that exactly 1 out of 4 consecutive days will be sunny?
C)What is the probability that at least 1 out of 4 consecutive days will be sunny?
I think part A is a Geometric distribution where a success is a sunny day and the number of trials is 3.
asked 2022-07-18
Probability of picking 2 numbers between 0 and 1 to be within 1/2 distance of each other?
What's the probability of picking 2 numbers, x and y, between 0 and 1 such that they will be within the distance of 12 of each other?
In other words, Pr ( distance between x and y 1 2 ) = ?
I solved the problem through a geometric approach by rewriting the probability as Pr ( | x y | 1 2 ) and graphing | x y | 1 2
From the graph, I calculated the red area to be 75%.
Question: What would be a non-geometric solution to this problem?

New questions