Zadanie 1a
Initially w=3, p=1/3 and charge is 1. (i) Show that the probability a of still running with charge 1 after exactly two ticks is 2/9. (ii) Find the probability b that full charge is reached for the first time on tick 4.
- Materiał graficzny
Combined-event diagrams and probabilities
Use tables, trees or Venn diagrams to reason about combined, conditional and independent events with or without replacement.
Tę umiejętność sprawdza 37 zadań wzorowanych na arkuszach IB z lat 2022–2026.
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Zadanie 1c
Now let w=4 and p=1/3. Express R₂ in terms of R₁ and R₃.
Zadanie 1d
For w=4 and p=1/3 it is known that R₁=1/15 and R₃=7/15. Find R₂.
Zadanie 1h
The upward-tick probability is now 0.498. A run starts with m charge units and stops successfully when it has accumulated 12 additional units, so capacity is w=m+12. Find the smallest positive integer m giving at least an 85% chance of success.
Zadanie 5a
Show that P(A) = (17 − x)/20.
Zadanie 7a
Find the probability that a randomly selected component triggers an alert.
Zadanie 10c
For type S, P(S<200)=0.30854 and P(200<S<224)=0.62465. Independently select one battery of each type. Find the probability that at least one lasts more than 224 minutes.
Zadanie 1d
Show that , explaining the probability calculation for every coefficient.
Zadanie 1e
(i) Find p. (ii) Find GY(t).
Zadanie 4b
Show that C and D are independent.
Zadanie 4a
Calculate P(B).
Zadanie 11c
B and C have error probabilities 0.04 and 0.12. Of all images, 50% go to A, 30% to B and 20% to C. For an image selected at random, find (i) the probability of an error; (ii) the probability it was processed by A, given an error.
Zadanie 11d
Before the update, B and C have error probabilities 0.04 and 0.12. Of all images, 50% go to A, 30% to B and 20% to C. After an update, C’s error probability is x%, while the other error rates and routing shares remain unchanged. The overall…
Zadanie 1a
For policy A with p=.12, find (i) the probability that the first inspection is on batch 4; (ii) the probability of at least one inspection among the first 8 batches.
Zadanie 1b
(i) Explain why P(X = x) = p(1 − p)x−1 for integer x≥1. (ii) Write E(X) in sigma notation.
Zadanie 1e
Show that under policy B the first inspection occurs on batch 3 with probability 405/1024.
Zadanie 1f
Explain why Y≤4.
Zadanie 4a
State the probability of no alert from (i) check A; (ii) check B.
Zadanie 4b
(i) Show that 16k2 − 40k + 9 = 0. (ii) The roots of this equation are 1/4 and 9/4. State the valid value of k and justify your choice.
Zadanie 6c
Find the probability that the first failure is the seventh transmission.
Zadanie 6b
A randomly selected pellet has mass greater than 16 g. Estimate the probability that it is type A. Give your answer as a fraction.
Zadanie 2b
Find P(A′ ∣ B′).
Zadanie 3
Events A,B satisfy P(A)=0.45, P(A|B)=0.30 and P(A∪B)=0.66. Find P(B).
Zadanie 7a
Complete the three blank branch probabilities in the tree.
- Materiał graficzny
Zadanie 11a
For experiment A, show that y2 − (4r + 1)y + r2 − r = 0.
Zadanie 11d
For experiment B, express P(B) in terms of y.
Zadanie 11e
One yellow counter is added to the container in experiment B. The probability of exactly one red and two yellow counters becomes 3/2 times its previous value. Find the original y.
Zadanie 2a
Find P(A∪B).
Zadanie 2b
Hence or otherwise find P(A′∩B′).
Zadanie 2a
Find P(A∪B).
Zadanie 2b
Hence or otherwise find P(A′∩B′).
Zadanie 3
Independent events A,B satisfy P(A)=2P(B) and P(A∪B)=0.72. Find P(B).
Zadanie 10c
After three cycles the recorded total is 2 units. Find the probability the batch passes.
Zadanie 6a
Find P(A∩B) if the events are independent.
Zadanie 6b
Find the smallest possible value of P(A∩B).
Zadanie 6c
Find the largest possible value of P(A∩B), with a justification.
Zadanie 10b
Four cycles are checked in order. Find the probability that only the fourth lasts longer than 48 minutes.