Answer:
The correct option is:
P(flower is hibiscus | color is red)=P(flower is hibiscus)
Step-by-step explanation:
We are given a table as:
Type of Flower/Color Red Pink Yellow Total
Rose 40 20 45 105
Hibiscus 80 40 90 210
Total 120 60 135 315
A)
P(flower is yellow|flower is rose)≠P(flower is yellow)
We know that:
P(A|B) is given by:
[tex]P(A|B)=\dfrac{P(A)\bigcap B)}{P(B)}[/tex]
Here let A= Flower is yellow.
B= flower is rose.
A∩B=flower is a yellow rose.
P(A∩B)=45/315=1/7
P(B)=105/315=1/3
P(A)=135/315=3/7
Hence,
[tex]P(A|B)=\dfrac{\dfrac{1}{7}}{\dfrac{1}{3}}\\\\\\P(A|B)=\dfrac{3}{7}[/tex]
Hence, we have:
[tex]P(A|B)=P(A)[/tex]
This means that:
P(flower is yellow|flower is rose)=P(flower is yellow)
Hence, option: A is incorrect.
B)
P(flower is hibiscus | color is red)=P(flower is hibiscus)
Let A=flower is hibiscus.
B=color is red.
A∩B=flower is red hibiscus.
Hence,
P(A)=2/3
P(B)=8/21
P(A∩B)=16/63
[tex]P(A|B)=\dfrac{\dfrac{16}{63}}{\dfrac{8}{21}}\\\\\\P(A|B)=\dfrac{2}{3}=P(A)[/tex]
Hence, we get:
P(flower is hibiscus | color is red)=P(flower is hibiscus)
Hence, option: B is correct.
C)
P(flower is rose | color is red)=P( Flower is red)
Let A=flower is rose
B= color is red.
A∩B=flower is red rose
Hence,
P(A)=1/3
P(B)=8/21
P(A∩B)=8/63
Hence, we have:
[tex]P(A|B)=\dfrac{1}{3}\neq P(B)[/tex]
Hence, option: C is incorrect.
D)
P(flower is hibiscus | color is pink)≠P(flower is hibiscus)
Let A=flower is hibiscus.
B= color is pink.
A∩B=flower is pink hibiscus.
Hence,
P(A)=2/3
P(B)=4/21
P(A∩B)=8/63
[tex]P(A|B)=\dfrac{\dfrac{8}{63}}{\dfrac{4}{21}}\\\\\\P(A|B)=\dfrac{2}{3}=P(A)[/tex]
Hence, we get:
P(flower is hibiscus | color is red)=P(flower is hibiscus)
Hence, this option is incorrect.
