Ursula takes a bus to the station and then immediately gets on a train to work. Is the probability of her missing the train independent or dependent on her missing her bus?

Write an equation for P \left( A \text{ and } B \right) in terms of P \left( A \right) and P \left( B \right) given that the two events are independent.

The probability that Ursula, Jimmy and Bianca get permission to go on their school trip are 0.5, 0.7 and 0.3, respectively. Find the probability that at least one of them gets permission.

Consider P\left(A\cap B\right) = 0.2 and P\left(A\cap B' \right) = 0.3, where A and B are independent.

Find the value of:

P \left( A \right)

P \left( B \right)

P \left(A\cup B' \right)

The probability of two independent events, A and B are, P \left( A \right) = 0.5 and P \left( B \right) = 0.8. Determine the probability of:

Both A and B occurring.

Neither A nor B.

A or B or both.

B but not A.

A given that B occurs.

Applications

Students were asked what they are allergic to. The table shows the results:

	Allergic to nuts	Not allergic to nuts
Allergic to dairy	14	12
Not allergic to dairy	25	25

If a student is chosen at random, find the probability that:

The student is allergic to dairy.

The student has an allergy.

The student does not have an allergy.

The student is also allergic to dairy if he or she is allergic to nuts.

A sample of 20 students, A - T, from the same school were asked whether they had a haircut in the last month (H) and whether they had been sick in the last month (S).

The results are shown in the given tables:

Student	A	B	C	D	E	F	G	H	I	J
\text{Hair}	✓	✓	-	✓	-	✓	✓	-	-	✓
\text{Sick}	-	✓	-	✓	-	-	✓	✓	-	-

Student	K	L	M	N	O	P	Q	R	S	T
\text{Hair}	-	-	-	✓	-	✓	-	✓	-	-
\text{Sick}	✓	✓	✓	✓	-	-	✓	✓	-	✓

Find the probability that a student:

Had a haircut in the last month.

Had been sick in the last month.

iii

Was sick and had a haircut in the last month.

Find the value of P \left( H \right) \times P \left( S \right).

Does the data suggest that getting haircuts and being sick in the last month are independent or dependent?

A school consisting of 490 primary students and 490 secondary students offers everyone an optional end of year project to improve their grade. There are various types of projects they can do.

A summary of the number of student submissions are shown in the given table:

	Essay	Creative story	Art work	Other Project	Did not submit
Primary	24	74	49	34	309
Secondary	88	82	82	62	176

Find the probability of selecting a student that submitted a project.

Find the probability of selecting a student that submitted a project given that they are in a secondary year.

Does the data suggest that doing a project and being a secondary student are independent or dependent?

Find the probability that a submitted project is an essay.

Find the probability that a submitted project is an essay given that the project was submitted by a primary student.

Does the data suggest that if a student did hand in a project, that being a primary student and writing an essay is independent or dependent?

100 residents of a city were asked "Do you support the construction of the new train station?". The people questioned were also classified as either eastern or western residents of the city. The results are shown in the table:

Does the data suggest that support for the train station is independent of whether a resident lives in the east or the west?

	East	West	Total
Yes	47	23	70
No	28	2	30
Total	75	25	100

480 random people in Australia were surveyed, examining their carbon footprint and the city they lived in. The people were then categorised as living in either an urban (U) or regional location (R), and whether that person has a high carbon emission (H) or low carbon emission (L). The results are shown in the table:

Does the data suggest that the type of city a resident lives in is independent to their carbon emission?

	U	R	\text{Total}
H	48	72	120
L	144	216	360
\text{Total}	192	288	480

126 patients were tested for a gene, and then tested to how attractive they were to mosquitos. Patients could either be categorised as repellant (R), neutral (N) or attractive (A) to mosquitos, and either had the standard (S) or mutant (M) gene. The results are shown in the table:

Does the data suggest that having the standard gene (S) and having a neutral attraction to mosquitos (N) are independent?

	R	N	A
S	22	30	38
M	8	12	16
\text{Total}	30	42	54

A class of 13 students had their height, shoe size and 400\text{ m} race time collected into the following table:

Let T be the event that a randomly chosen student from the class has a race time greater than 90. Find the value of P \left( T \right).

Let H be the event that a randomly chosen student from the class has a height greater than 130. Find the value of P \left( H \right).

Find the value of P\left(T|H\right).

Let S be the event that a randomly chosen student from the class has a shoe size greater than or equal to a size 10. Find the value of P\left(H|S\right).

Hence, which of the following events appear to be dependent, H and S or \\ H and T?

\text{Athlete}	400 \text{ m} \\\ \text{time (sec)}	\text{Height} \\ \text{(cm)}	\text{Shoe} \\ \text{size}
\text{Aaron}	94	137	8
\text{Tobias}	62	165	10
\text{Lisa}	89	100	11
\text{Georgia}	64	155	7
\text{Ben}	98	100	10
\text{Patricia}	127	102	11
\text{Vanessa}	84	147	7
\text{Sandy}	98	144	8
\text{Adam}	54	154	7
\text{Amelia}	84	147	8
\text{Rosey}	81	126	12
\text{Neil}	104	134	9
\text{Roald}	67	127	8

A museum has new software to determine whether a painting is fake or authentic. The system has an accuracy of 75\%. Art experts have previously determined that 5\% of paintings that the museum examines are fakes.

Find the probability that:

The software correctly identifies a painting as fake.

The software incorrectly identifies an authentic painting as fake.

The software will identify a painting as fake.

A painting that is identified as fake is actually fake, correct to two decimal places.

8.03 Independent and dependent events

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