Question

Problem 7: A cloud of particles and anti-particles suddenly appears and immediately all parti- cle/anti-particle pairs are instantaneously annihilated. When the cloud forms, each bit of stuff has a probability of 0.5001 of being a particle and a probability of 0.4999 of being an anti-particle. Let n, a very large integer, be the number of particles/anti-particles at the start and for i = 1,2,...,n let: 5+1 if the i-th bit of stuff is a particle 1-1 if the i-th bit of stuff is an anti-particle. When the destruction completes, one of three states remains: there are only particles left, there are only anti-particles left, there is nothing left. Part 7a: Write an expression in terms of the X, that represents the final state. Part 7b: Show that the probability that only anti-particles remain after the annihilation process completes is approximately 0? Part 7c: On average, how many particles/anti-particles remain after annihilation? Problem 8: A spherical balloon of volume V and radius r (V = 473) is inflated with a volume of gas that follows a Normal distribution with mean = 3x cm and variance of ?. What is the probability that the diameter of the balloon will lie between 2 and 3 cm? (Leave your answer in terms of "(?).)

Answer 1

Solution : problema: (Fa) Expression interns of xi: Givere, a cloud of particles and anti particles suddenly appears and immediately all particle) anti particle pairs are instanteneously an inelated. At the time of formation of cloud, Probability of particle = 0.5ool probability of antiparticle: = 0.4999 stl; for particle t ; for anti-particle . let state for particle be 1+1> and state for anti particle be 1-13 > final state = cilt1> +21-1) 14>= cil+13 +621-130

where 1912+ 1 C2/² = 1 kit2 = probability for particle lot = probability for anti-particle Hence 1912= 0.500l Q = 10.5001 - 16212 = 0.4999 cz = 50.4999 - from 004 (14>= 50.8007/+1> +1 0.9999 1127 particle in cloud. where lys = state of Ab) The probability that only anti particles remain after the aninhilation process completes is approximately o: - when cloud annihilation then "0.50014"

Particle will combine with 110.4999h" antipaticles ce number of particle(lanti particles) left after annihilation con) = (noof particles) - (noof antiparticles on losooin – 0.499911 Ən (0.500 1-0.4999) -0.0002n] Probability of antiparticles ramain = in 0.00021 = 0.0002 Hence, no of antiparticles after the annihilation is zero (approx).

(7) Noof particles / antiparticles remain after the annihilation are: Noof particles = 0.50001-0:4999n = 0.00021 va . As the roof particles is more than the noof antiparticles, Hence affer Complete annihilation, on average no anti particles will be preſent.