Answer:
The name of the distribution of number of trials
X
X:
Geometric Distribution.
As the trials will be continued until the first success i.e draw of the heart is achieved.
b. The mean number of draws needed:
The probability of success or the probability of drawing a heart from the deck:
p
=
Number of hearts
Total
=
13
52
=
0.25
p=Number of heartsTotal=1352=0.25
The mean number of trials required to get the first success:
μ
=
1
p
=
1
0.25
=
4
μ=1p=10.25=4
c. The standard deviation of
X
X:
The standard deviation of geometric distribution:
σ
=
√
1
−
p
p
2
=
√
1
−
0.25
0.25
2
≈
3.464
σ=1−pp2=1−0.250.252≈3.464
d. The probability distribution table:
The probability mass function of geometric distribution:
P
(
X
=
r
)
=
(
1
−
p
)
n
⋅
p
P(X=r)=(1−p)n⋅p
where
X
∈
[
0
,
1
,
2
,
.
.
.
,
n
]
X∈[0,1,2,...,n]