Conditional Probability — Basic

Conditional Probability — Basic. Practice questions to deepen understanding of basic conditional probability. Online math practice with full solutions and step-by-step explanations.

Basic conditional probability practice — definition of conditional probability, Bayes’ formula, probability trees. Explanations from the basics for beginners.

40 questions

Question 1

2.50 pts

🎯 probability ?
probability :

probability that an event occurs given another event already occurred

probability

probability event

probability event No

Explanation:

💡 :

1: 🔍

Question 2

2.50 pts

✏️ :
P(A|B) :

probability -A given -B

P(A and B)

P(A or B)

probability -B given A

Explanation:

💡 :

🔍

probability 🎯

probability event A
given event B

: P(A|B)

: "P A given B"

P(A|B)

because | :
"given"

A =
B =

! 📐

Question 3

2.50 pts

📐 :
probability ?

P(A|B) = P(A∩B) / P(B)

P(A|B) = P(A) × P(B)

P(A|B) = P(A) + P(B)

P(A|B) = P(A) / P(B)

Explanation:

💡 :

🔍

P(A|B) = P(A∩B) / P(B)

? 📐

P(A∩B) = probability

P(B)

Question 4
2.50 pts

🌧️ Example:
"The probability that it is cold, given that it is raining"
How is this denoted?

P(Cold | Rain)

P(Rain | Cold)

P(Cold and Rain)

P(Cold) × P(Rain)

Explanation:

💡 Detailed explanation:
Sentence analysis 🔍
"The probability that it is cold,
given that it is raining"

A = Cold (what we want)
B = Rain (what we know)

P(Cold | Rain)
Answer: P(Cold | Rain)

Question 5
2.50 pts

🔄 :
P(A|B) = P(B|A)?

No - !

Yes - always equal

Yes - because

No

Explanation:

💡 :
🔍
! ⚠️
P(A|B) ≠ P(B|A)

!
📐

Question 6
2.50 pts

🔬 :
calculate P(A|B), :

sample event -B

enlarging the sample space

-B

A -B

Explanation:

💡 :
🔍
sample! 🔬
-B ,

B No !

B ""
: sample event -B

Question 7
2.50 pts

∩ Notation:
What is P(A∩B)?

The probability that both A and B occur

The probability that A or B occurs

The probability of A given B

The probability that A does not occur

Explanation:

💡 Detailed explanation:
Explanation 🔍
A ∩ B
∩ = intersection = "and"

P(A∩B) = the probability
that both occur
Answer: The probability that both A and B occur

Question 8
2.50 pts

✖️ :
P(A|B) = P(A∩B)/P(B)
:

P(A∩B) = P(A|B) × P(B)

P(A∩B) = P(A|B) + P(B)

P(A∩B) = P(A|B) / P(B)

P(A∩B) = P(A) + P(B)

Explanation:

💡 :
🔍
P(A|B) = P(A∩B) / P(B)

-P(B):

P(A∩B) = P(A|B) × P(B)
📐

Question 9
2.50 pts

📊 because:
probability P(A|B) :

0 -1 ()

0 1

any positive number

only whole numbers

Explanation:

💡 :
🔍
probability!
0 ≤ P(A|B) ≤ 1

• P(A|B) = 0 → A -B
• P(A|B) = 1 → A -B
: 0 -1 ()

Question 10
2.50 pts

🎲 :
.
known (2,4,6).
probability 4?

1/3

1/6

1/2

2/3

Explanation:

Explanation: See the relevant definition and formula above.

Question 11
2.50 pts

🎲 :
.
known large -3 (4,5,6).
probability ?

2/3

1/2

1/3

1/6

Explanation:

💡 :
1: 🔍
known: large -3

: {4, 5, 6}

3
2: ? 📐
{4, 5, 6}:

:

Question 12
2.50 pts

🃏 :
(52 ).
known (/).
probability ?

1/2

1/4

13/52

1/13

Explanation:

Explanation: See the relevant definition and formula above.

Question 13
2.50 pts

📐 :
given: P(A∩B) = 0.2, P(B) = 0.5.
P(A|B)?

0.4

0.7

0.1

0.25

Explanation:

💡 :
1: 🔍
P(A|B) = P(A∩B) / P(B)
2: 📐
P(A|B) = 0.2 / 0.5

= 2/5

=

Question 14
2.50 pts

📐 :
given: P(A∩B) = 0.3, P(B) = 0.6.
P(A|B)?

0.5

0.9

0.3

0.18

Explanation:

💡 :
📐
P(A|B) = P(A∩B) / P(B)

= 0.3 / 0.6

= 0.5
: 0.5

Question 15
2.50 pts

📐 :
given: P(A|B) = 0.4, P(B) = 0.5.
P(A∩B)?

0.2

0.9

0.8

0.1

Explanation:

💡 :
1: 🔍
P(A∩B) = P(A|B) × P(B)
2: 📐
P(A∩B) = 0.4 × 0.5

=

Question 16
2.50 pts

📊 :
because 30 .
20 , 15 .
10 .
P( | )?

1/2

1/3

2/3

1/4

Explanation:

💡 :
1: 🔍
🔹 : 20
🔹 : 10

: P( | )
2: 📐
P( | )

= P() / P()

= (10/30) / (20/30)

= 10/20
3: 💭

Question 17
2.50 pts

🪙🪙 :
2 .
known "".
probability ""?

1/3

1/2

1/4

2/3

Explanation:

💡 :
1: sample No 🔍
:
-, -, -, -

4
2: 📐
" " =

-, -, -

3 <

Question 18
2.50 pts

🔴🔵 :
: 3 , 2 .
.
known No .
probability ?

1

3/5

2/5

1/2

Explanation:

💡 :
1: 🔍
: 3 , 2

known: No
Yes: !
2: 📐
No ,
!

P( | No ) = 1

Question 19
2.50 pts

📐 :
given: P(A|B) = 0.5, P(A∩B) = 0.3.
P(B)?

0.6

0.8

0.15

0.2

Explanation:

💡 :
1: 🔍
P(A|B) = P(A∩B) / P(B)

:

P(B) = P(A∩B) / P(A|B)
2: 📐
P(B) = 0.3 /

Question 20
2.50 pts

🎲🎲 :
2 .
known 7.
probability 3?

1/6

1/36

1/7

2/6

Explanation:

💡 :
1: = 7? 🔍
7:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

6
2: = 3? 📐

Question 21
2.50 pts

🔗 -:
event A -B dependent :

P(A|B) = P(A)

P(A|B) = P(B)

P(A|B) = 0

P(A|B) = 1

Explanation:

Explanation: See the relevant definition and formula above.

Question 22
2.50 pts

🎲🪙 :
.
A = " 6"
B = " "
dependent?

No - dependent

Yes - dependent

depends on the outcome

cannot be determined

Explanation:

💡 :
🔍
dependent! ✓

No

!

: No - dependent

Question 23
2.50 pts

🌧️ :
A = " "
B = " "
dependent?

Yes - dependent

No - dependent

depends on the weather

cannot be determined

Explanation:

💡 :
🔍
dependent! 🔗
,
!

P( | ) > P()

Yes dependent
: Yes - dependent

Question 24
2.50 pts

🔢 :
A -B dependent.
P(A) = 0.3, P(B) = 0.4.
P(A∩B)?

0.12

0.7

0.1

0.58

Explanation:

💡 :
1: 🔍
event dependent:

P(A∩B) = P(A) × P(B)
2: 📐
P(A∩B) = 0.3

Question 25
2.50 pts

🔍 :
P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2.
A -B dependent?

Yes - because P(A)×P(B) = P(A∩B)

No - No

cannot be determined

depends on the context

Explanation:

💡 :
1: 🔍
dependent :
P(A) × P(B) = P(A∩B)
2: 📐
P(A) × P(B) = 0.5 × 0.4 = 0.2

P(

Question 26
2.50 pts

🔍 :
P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.3.
A -B dependent?

No - because P(A)×P(B) ≠ P(A∩B)

Yes - always dependent

cannot be determined

Yes -

Explanation:

💡 :
1: 🔍
P(A) × P(B) = 0.5 × 0.4 = 0.2

P(A∩B) = 0.3

0.2 ≠ 0.3 ✗
2: 📐

Question 27
2.50 pts

🔢 :
A -B dependent.
P(A) = 0.6.
P(A|B)?

0.6

-P(B)

0.4

1

Explanation:

💡 :
🔍
-! 💡
dependent:

P(A|B) = P(A)

Yes: P(A|B) = 0.6
: 0.6

Question 28
2.50 pts

⛓️ :
3 .
.
probability -3 ""?

1/8

1/2

3/8

1/4

Explanation:

💡 :
1: - 🔍

Yes :
2: 📐
P(,,) = P() × P() × P()

= 1/2 × 1/2 × 1/2

= <

Question 29
2.50 pts

🔴🔵 No :
: 3 , 2 .
2 No .
?

No -

Yes - always

depends on the first ball

Yes - event

Explanation:

💡 :
🔍
! 🔗
No = !

,
!

probability
: No -

Question 30
2.50 pts

🔴🔵 :
: 3 , 2 .
2 .
?

Yes -

No -

depends on the ball

cannot be determined

Explanation:

💡 :
🔍
! ✓
= -!

=
: Yes -

Question 31
2.50 pts

🏥 :
1% population .
Correct -90%.
P( | )?

0.9 (90%)

0.01 (1%)

0.1 (10%)

0.99 (99%)

Explanation:

💡 :
🔍
:

" Correct -90%"

P( + | )!
📐
P( +

Question 32
2.50 pts

⚽ :
60% .
40% .
probability ?

0.24

0.60

0.40

1.00

Explanation:

💡 :
1: 🔍
P( | ) = 0.6
P() = 0.4

: P( ∩ )
2: 📐
P( ∩ ) = P(|) × P()

= 0.6 × 0.4

= 0.24

Question 33
2.50 pts

📚 :
70% .
, 80% .
P( ∩ )?

0.56

0.80

0.70

1.50

Explanation:

Explanation: See the relevant definition and formula above.

Question 34
2.50 pts

☀️🌧️ :
probability = 0.3.
, probability = 0.8.
P( )?

0.24

0.80

0.30

1.10

Explanation:

Explanation: See the relevant definition and formula above.

Question 35
2.50 pts

🔴🔵 :
4 , 3 .
2 No .
P( | )?

3/6 = 1/2

4/7

3/7

4/6

Explanation:

💡 :
1: 🔍

:
3 , 3

": 6
2: 📐
P( | )

= 3/6 =

Question 36
2.50 pts

🔴🔴 :
4 , 3 .
2 No .
P( )?

12/42 = 2/7

16/49

4/7

1/2

Explanation:

💡 :
1: 🔍
P( )

= P( ) × P( | )
2: 📐
= 4/7 × 3/6

= 4/7 × 1/2

= 4/14 =

Question 37
2.50 pts

🌳 probability:
probability calculate:

See explanation

mean/average

standard deviations only

simple probabilities only

Explanation:

💡 :
🔍
probability 🌳
No
calculate

📍 = event
📍 No
📍
:

Question 38
2.50 pts

🍕 :
60% .
, 30% .
.
probability ?

0.30

0.18

0.60

0.90

Explanation:

💡 :
🔍
:

" ,
30% "

= P( | ) = 0.30
📐
<

Question 39
2.50 pts

🎰 :
10 1-10.
.
known (2,4,6,8,10).
P(large -7 | )?

2/5

3/10

1/2

2/10

Explanation:

💡 :
1: 🔍
known:

: {2, 4, 6, 8, 10}

5
2: large -7 📐
{2, 4, 6, 8, 10}:

Question 40
2.50 pts

🌟 :
Correct probability ?

See explanation

P(A|B) = P(A∩B) / P(B)

P(A|B) ≠ P(B|A)

dependent P(A|B) = P(A)

Explanation:

💡 :
🔍
probability - ! 🎯
📐
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