350 tourists, Visited Japan (J), 157 ed Korea (K), Singapore (S), 59 visited Japan and Korea, 81 visited Japan and Singapore, 94 visited Korea and Singapore and 37 visited Korea, Japan, and Singapore. If a tourist is randomly chosen from this group, what is the probability of choosing a tourist who visited a) Japan? b) Japan only? c) Korea or Singapore? d) Japan and Singapore? e) Singapore but not Korea?
To solve this problem, we can use a Venn diagram to visualize the information given:
“`
J
/ \
/ \
K S
|\ /|
| \ / |
| X |
| / \ |
|/ \|
J K
S
“`
where J represents the set of tourists who visited Japan, K represents the set of tourists who visited Korea, and S represents the set of tourists who visited Singapore. The numbers inside the regions represent the number of tourists who visited each combination of countries.
a) To find the probability of choosing a tourist who visited Japan, we need to add up the numbers in the J region and divide by the total number of tourists:
P(J) = (J + JK + JS) / (J + K + S + JK + JS + KS + JKS)
= (350 + 59 + 81) / (350 + 157 + 59 + 81 + 94 + 37 + 1)
= 490 / 739
≈ 0.663
Therefore, the probability of choosing a tourist who visited Japan is approximately 0.663.
b) To find the probability of choosing a tourist who visited Japan only, we need to look at the J region that is not shared with any other region:
P(J only) = J / (J + JK + JS)
= 350 / (350 + 59 + 81)
= 350 / 490
= 0.714
Therefore, the probability of choosing a tourist who visited Japan only is 0.714.
c) To find the probability of choosing a tourist who visited Korea or Singapore, we need to add up the numbers in the K and S regions and subtract the number in the KS region (to avoid double-counting):
P(K or S) = (K + KS + S + JS) / (J + K + S + JK + JS + KS + JKS)
= (157 + 59 + 94 + 81) / (350 + 157 + 59 + 81 + 94 + 37 + 1)
= 391 / 739
≈ 0.529
Therefore, the probability of choosing a tourist who visited Korea or Singapore is approximately 0.529.
d) To find the probability of choosing a tourist who visited Japan and Singapore, we need to look at the JS region:
P(JS) = JS / (J + K + S + JK + JS + KS + JKS)
= 81 / 739
≈ 0.110
Therefore, the probability of choosing a tourist who visited Japan and Singapore is approximately 0.110.
e) To find the probability of choosing a tourist who visited Singapore but not Korea, we need to subtract the number of tourists who visited both Singapore and Korea (KS) from the number of tourists who visited Singapore (S):
P(S but not K) = (S – KS) / (J + K + S + JK + JS + KS + JKS)
= (94 – 37) / 739
≈ 0.063
Therefore, the probability of choosing a tourist who visited Singapore but not Korea is approximately 0.063.
