Probability of Combined Events (Copy)
1.
Two coins are tossed. Write the sample space. Find P(2 Heads).
2.
From Q1, find P(at least one Tail).
3.
A die is rolled twice. Find P(getting a total of 7).
4.
A bag has 3 red and 2 blue balls. Two balls drawn without replacement. Find P(both red).
5.
From Q4, find P(1 red and 1 blue).
6.
From Q4, find P(both same colour).
7.
A bag has 5 green and 3 yellow counters. Two counters drawn with replacement. Find P(both green).
8.
From Q7, find P(one green and one yellow).
9.
From Q7, find P(both yellow).
10.
In a class, 60% like Maths, 50% like Science, and 30% like both. Represent with a Venn diagram. Find P(likes Maths or Science).
11.
From Q10, find P(likes neither subject).
12.
A bag has 4 black and 6 white balls. One is picked at random, replaced, then another picked. Find P(1 black and 1 white).
13.
From Q12, find P(both black).
14.
A card is drawn from a deck. Find P(spade or king).
15.
From Q14, find P(spade and king).
16.
Two dice rolled. Find P(both even).
17.
From Q16, find P(sum = 9).
18.
From Q16, find P(at least one 6).
19.
A coin tossed and a die rolled. Find P(head and even number).
20.
From Q19, find P(tail or odd number).
21.
Bag contains 2 red, 3 blue, 5 green. Two balls chosen without replacement. Find P(both green).
22.
From Q21, find P(2 different colours).
23.
In a survey, 40% watch football, 30% watch cricket, 15% watch both. Find P(football ∪ cricket).
24.
From Q23, find P(neither).
25.
A die is rolled twice. Find P(first is 4, second is even).
26.
From Q25, find P(both rolls are ≤ 3).
27.
Bag contains 10 sweets: 4 chocolate, 6 fruit. Two chosen without replacement. Find P(1 chocolate and 1 fruit).
28.
From Q27, find P(both chocolate).
29.
From Q27, find P(both fruit).
30.
Two cards drawn without replacement from a pack of 52. Find P(both aces).
1.
Sample space = {HH, HT, TH, TT}.
P(2 Heads) = 1/4.
2.
At least one Tail = {HT, TH, TT}.
P = 3/4.
3.
Possible pairs for total 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1).
6 outcomes out of 36.
P = 6/36 = 1/6.
4.
First red = 3/5, then red = 2/4.
P = (3/5)×(2/4) = 6/20 = 3/10.
5.
(3/5×2/4)+(2/5×3/4) = 6/20+6/20=12/20=3/5.
6.
Both same colour = P(2 red)+P(2 blue).
= 3/10 + (2/5×1/4)=3/10+2/20=3/10+1/10=4/10=2/5.
7.
With replacement: P(green)=5/8 each.
P(both green) = (5/8)²=25/64.
8.
One green and one yellow = 2×(5/8×3/8)=30/64=15/32.
9.
P(both yellow)=(3/8)²=9/64.
10.
P(Maths ∪ Science) = 0.6+0.5−0.3=0.8.
11.
P(neither)=1−0.8=0.2.
12.
P(black then white)=(4/10×6/10)=24/100.
P(white then black)=(6/10×4/10)=24/100.
Total=48/100=12/25.
13.
P(both black)=(4/10)²=16/100=4/25.
14.
P(spade)=13/52=1/4.
P(king)=4/52=1/13.
Double counted king of spades once → subtract 1/52.
Total=1/4+1/13−1/52=16/52=4/13.
15.
P(spade ∩ king)=1/52.
16.
Each die: 3 evens out of 6 → 1/2.
P(both even)=(1/2)²=1/4.
17.
Sum=9 pairs: (3,6),(4,5),(5,4),(6,3). 4 outcomes.
P=4/36=1/9.
18.
At least one 6 = 1 − P(no 6).
=1−(5/6×5/6)=1−25/36=11/36.
19.
Coin Head (1/2), Die even (3/6=1/2).
P=1/2×1/2=1/4.
20.
Tail or odd.
P(tail)=1/2. P(odd)=1/2.
But overlap tail+odd=1/4.
So P=1/2+1/2−1/4=3/4.
21.
Total=10. Greens=5.
Without replacement: (5/10×4/9)=20/90=2/9.
22.
2 different colours = 1−P(both same).
P(both same)=(2/10×1/9)+(3/10×2/9)+(5/10×4/9)=2/90+6/90+20/90=28/90=14/45.
So P(diff)=31/45.
23.
P(F ∪ C)=0.4+0.3−0.15=0.55.
24.
P(neither)=1−0.55=0.45.
25.
First 4:1/6. Second even:3/6=1/2.
P=1/12.
26.
Both ≤3. P=3/6 for each=1/2.
So P=1/4.
27.
P(choc then fruit)=(4/10×6/9)=24/90.
P(fruit then choc)=(6/10×4/9)=24/90.
Total=48/90=8/15.
28.
P(both choc)=(4/10×3/9)=12/90=2/15.
29.
P(both fruit)=(6/10×5/9)=30/90=1/3.
30.
P(first ace)=4/52=1/13.
P(second ace)=3/51.
Total=1/13×3/51=3/663=1/221.