Probability and Reliability

1. Probability and Reliability Thirteen Meeting

2. Probability: Occurrence • Process has more than one outcome • Occurrence of event Ei • is the number of times, Ni, that Ei occurs out of a total number, N, of events. • r = number of events • Example: • E1 = picking a red ball • E2 = picking a greenball • E3 = picking a blackball • N1 = 2 • N2 = 3 • N3 = 2 • N= N1 + N2 +N3 = 7 • r= 3

3. Probability: Relative Frequency or Probability • It is the frequency of occurrence of an event • It is the frequency of occurrence of an event Ei • N = N1+ N2 + …+ Nr-1 + Nr • P(E1) + P(E2) + …+ P(Er-1) + P(Er) = 1

4. Probability: Probability Example • r = 3 • E1= Picking a red ball • E2= Picking a green ball • E3= Picking a black ball • N1 = 2 • N2 = 3 • N3 = 2 • N = 2 + 3 + 2 = 7 • P(E1) = 2/7 • P(E2) = 3/7 • P(E3) = 2/7 • 2/7 + 3/7 + 2/7 = 1

5. Venn Diagram • S = The total area covered by a Venn diagram represents all the events • E = The frequency of a specific event corresponds to a given area of the diagram • The ratio (area of E)/(area of S) = the probability of the event E occurring.

6. No Occurrence Event • P(E) is the probability that event E occurs • P( Ē ) is the probability that the event does not occur • P( E ) + P( Ē ) = 1 • Example • E1= Picking a red ball • N1 = 2 • N = 7 • P(E1) = 2/7 • P(Ē1) = 5/7

7. S S B S P(B) A A or B B B A AB A And/Or Probabilities • P(AB) (joint probability) is the Probability that events A and B will both occur. • What is P(AB) if A and B Mutual Exclusive • P(AB) = 0 • P(A OR B) is the probability that event A, or event B, or both event A and event B will occur. • P( A OR B ) = P( A ) + P( B) - P(AB)

8. S = 100 S S = 100 B = 20 B = 20 B = 30 A=10 A = 25 A = 25 Examples P( A OR B ) = P( A ) + P( B) – P(AB) P( A OR B ) = 10/100 + 30/100 – 10/100 = 30/100 S = 100 B = 30 A=10 P( A OR B ) = P( A ) + P( B) – P(AB) P( A OR B ) = 25/100 + 20/100 – 0/100 = 45/100

9. Conditional Probabilities S • P( A | B) • A occur if B has definitely occurred. • P( A | B) = P(AB)/P(B) • Example: • P(T|C) = P(TC)/P(C) • P(T|C) = 0.08/0.4 = 0.2 T C CT

10. Independent Probabilities • Independent events • The occurrence of one event does not affect the probability of occurrence of the other. • P( A | B) = P( A) for independent events A and B • Example • Are events C and T independent • P(T) = P(T|C) must be true • P(T|C) = P(TC)/P(C) • P(T|C) = 0.08/0.4 = 0.2 • P(T) = 0.15 • The answer is NO C T CT

11. Independent Probabilities (cont.) • Example • Are events C and T independent • P(T) = P(T|C) must be true • P(T|C) = P(TC)/P(C) • P(T|C) = 0.06/0.4 = 0.15 • P(T) = 0.15 • The answer is YES

12. Multiplication Probabilities • P(AB) = P( A|B) P( B) • If A and B are independent events, • P(AB) = P( A) P( B) • Example: • P(TC) = P( T|C) P( C) • P(T|C) = P(TC)/P(C) • P(T|C) = 0.08/0.4 = 0.2 • P(TC) = 0.2 * 0.15 • P(TC) = 0.03

13. Binomial Distribution • Employee receives phone calls • 40% of phone calls = Cell phone (p) • 60% of phone calls = Ground phone (q) • Event that 2 out 3 = Cell phone (r) • What is P(r) • p = 0.4 • q = 0.6 • n = 3 • P(r) = ? 6! = 6 * 5 *4 * 3 * 2 * 1

14. Binomial Distribution • You send 10 postcards from 10 different towns • 1% of mail is lost • What is the probability that 3 of 10 postcards won’t arrive • Lost mail = • 0.01 = p • Received mail = • 0.99 = q • Total number of events = • 10 = n • Event 3 of 10 are lost = • 3 = r

15. Binomial Coefficient • nCr is Binomial coefficient

16. Binomial Coefficient Row 2 corresponds to 1C0 and 1C1. Row 3 corresponds to 2C0 , 2C1, and 2C2. What 10C7 = 120

17. Reliability Function • R(t), is the probability that a component will survive to time t. • This can be estimated by a test of N0 components: • Ns(t) is the number of components surviving at time t.

18. Reliability Function • Q(t), is the probability that a component will fail to time t. • This can be estimated by a test of N0 components: • Nf(t) is the number of components failing at time t.

19. Failure Density Function • ƒ(t). The probability of failing, ΔQ(t), during a very small interval Δt at t can be expressed as: • ΔQ(t) = ƒ(t) Δt

20. Continuous Function • Reliability function R(t) and failure function Q(t), the relationship to the failure density function ƒ(t) can be found by differentiation: • Equivalently, Q(t) and R(t) can be found by integrating ƒ(t):

21. Reliability Function Definitions • This is known as the normalization condition. • R(t) and Q(t) is called • cumulative probability distributions • because they give the probability of something happening up to a specified time. • ƒ(t) is called: • probability density function • because it is used to give the probability of failure during an interval of time • h(t) is called: • hazard rate, or failure rate • is the ratio of the number of failures per unit time, at time t, to the number of components exposed to failure at that time:

22. Hazard Rate Plot • Burn-in period (early life ) • A steep fall in the and it is when faulty components are weeded out), • Useful life period (or normal operating period) • a level when h(t) is effectively constant, • Wear-out (or end-of-life period) • when the hazard rate rises steeply. • The hazard rate is also related to the failure density function and the reliability function by the equation:

23. Exponential Probability Distribution • When h is constant, the R(t) and f(t) are simply: • R(t ) = exp(-ht ) = e-ht • f (t ) = h exp(−ht ) = h*e-ht

24. Reliability Function for Constant Hazard Rate h

25. Failure Density for Constant Hazard Rate h

26. Average Lifetimes: MTTF • Mean Time To Failure (MTTF) • The average lifetime of a batch of components • Example • if N1 failed at time t1, N2 failed at time t2, …etc., then: • If h is constant, then 10d=5,20d=6,30=4 50+120+120/15

27. Average Lifetimes: MTBF, MTTR and availability • Mean Time Between Failure (MTBF) • Is used for repairable components (no MTTF) • Is the average time from when a component starts to be operational until it fails • Mean Time To Repair (MTTR) • is the mean time taken to repair a fault. • Availability, A: