Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
10. Car

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 325 times and lost 675 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Goat
10. Car
11. Goat
12. Goat
13. Car
14. Goat
15. Car
16. Goat
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Goat
24. Goat
25. Goat
26. Car
27. Goat
28. Car
29. Goat
30. Goat
31. Goat
32. Goat
33. Goat
34. Goat
35. Goat
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Goat
46. Car
47. Goat
48. Car
49. Car
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Car
56. Car
57. Car
58. Car
59. Goat
60. Goat
61. Goat
62. Goat
63. Car
64. Car
65. Goat
66. Goat
67. Goat
68. Goat
69. Car
70. Goat
71. Goat
72. Car
73. Goat
74. Car
75. Goat
76. Car
77. Car
78. Car
79. Goat
80. Car
81. Goat
82. Goat
83. Car
84. Goat
85. Goat
86. Goat
87. Car
88. Goat
89. Goat
90. Car
91. Goat
92. Goat
93. Goat
94. Goat
95. Goat
96. Car
97. Car
98. Goat
99. Goat
100. Goat
101. Car
102. Car
103. Goat
104. Goat
105. Goat
106. Goat
107. Car
108. Car
109. Goat
110. Goat
111. Car
112. Goat
113. Goat
114. Goat
115. Goat
116. Goat
117. Goat
118. Goat
119. Goat
120. Goat
121. Car
122. Goat
123. Goat
124. Goat
125. Goat
126. Car
127. Car
128. Goat
129. Goat
130. Car
131. Goat
132. Goat
133. Goat
134. Goat
135. Goat
136. Car
137. Goat
138. Goat
139. Goat
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Goat
146. Goat
147. Car
148. Goat
149. Goat
150. Car
151. Goat
152. Car
153. Car
154. Goat
155. Car
156. Goat
157. Goat
158. Goat
159. Car
160. Goat
161. Car
162. Goat
163. Goat
164. Car
165. Car
166. Goat
167. Goat
168. Goat
169. Car
170. Car
171. Goat
172. Goat
173. Goat
174. Car
175. Goat
176. Goat
177. Goat
178. Car
179. Goat
180. Car
181. Goat
182. Goat
183. Car
184. Goat
185. Goat
186. Goat
187. Goat
188. Goat
189. Goat
190. Car
191. Car
192. Car
193. Car
194. Goat
195. Car
196. Goat
197. Goat
198. Car
199. Car
200. Car
201. Goat
202. Goat
203. Goat
204. Goat
205. Goat
206. Goat
207. Goat
208. Car
209. Car
210. Car
211. Car
212. Car
213. Car
214. Goat
215. Goat
216. Goat
217. Goat
218. Car
219. Car
220. Goat
221. Goat
222. Car
223. Goat
224. Car
225. Goat
226. Car
227. Goat
228. Goat
229. Goat
230. Goat
231. Goat
232. Car
233. Goat
234. Goat
235. Car
236. Goat
237. Car
238. Goat
239. Goat
240. Goat
241. Car
242. Goat
243. Car
244. Goat
245. Car
246. Goat
247. Goat
248. Goat
249. Goat
250. Goat
251. Car
252. Car
253. Car
254. Goat
255. Goat
256. Goat
257. Car
258. Goat
259. Goat
260. Goat
261. Goat
262. Car
263. Goat
264. Goat
265. Goat
266. Goat
267. Goat
268. Car
269. Goat
270. Goat
271. Goat
272. Goat
273. Goat
274. Goat
275. Car
276. Goat
277. Car
278. Goat
279. Goat
280. Goat
281. Goat
282. Car
283. Goat
284. Goat
285. Goat
286. Goat
287. Goat
288. Goat
289. Car
290. Car
291. Goat
292. Car
293. Car
294. Goat
295. Car
296. Goat
297. Car
298. Goat
299. Goat
300. Goat
301. Goat
302. Car
303. Car
304. Goat
305. Car
306. Car
307. Goat
308. Car
309. Goat
310. Car
311. Goat
312. Goat
313. Goat
314. Goat
315. Car
316. Car
317. Goat
318. Car
319. Goat
320. Goat
321. Goat
322. Goat
323. Goat
324. Goat
325. Goat
326. Goat
327. Goat
328. Goat
329. Goat
330. Car
331. Goat
332. Goat
333. Goat
334. Car
335. Car
336. Goat
337. Goat
338. Goat
339. Goat
340. Goat
341. Goat
342. Car
343. Car
344. Goat
345. Goat
346. Goat
347. Goat
348. Car
349. Goat
350. Car
351. Goat
352. Goat
353. Goat
354. Car
355. Goat
356. Goat
357. Car
358. Goat
359. Goat
360. Goat
361. Goat
362. Car
363. Goat
364. Goat
365. Car
366. Car
367. Goat
368. Car
369. Car
370. Car
371. Car
372. Goat
373. Goat
374. Goat
375. Goat
376. Goat
377. Goat
378. Car
379. Goat
380. Goat
381. Car
382. Goat
383. Goat
384. Goat
385. Car
386. Goat
387. Goat
388. Goat
389. Goat
390. Car
391. Goat
392. Goat
393. Car
394. Goat
395. Goat
396. Car
397. Car
398. Goat
399. Goat
400. Car
401. Car
402. Goat
403. Goat
404. Goat
405. Goat
406. Car
407. Car
408. Goat
409. Goat
410. Goat
411. Car
412. Goat
413. Car
414. Car
415. Goat
416. Goat
417. Goat
418. Goat
419. Goat
420. Car
421. Car
422. Goat
423. Goat
424. Goat
425. Goat
426. Car
427. Goat
428. Goat
429. Goat
430. Goat
431. Goat
432. Car
433. Car
434. Car
435. Goat
436. Goat
437. Goat
438. Goat
439. Goat
440. Goat
441. Goat
442. Goat
443. Goat
444. Goat
445. Goat
446. Goat
447. Goat
448. Car
449. Goat
450. Goat
451. Goat
452. Car
453. Goat
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Goat
460. Car
461. Car
462. Goat
463. Goat
464. Car
465. Goat
466. Goat
467. Car
468. Goat
469. Goat
470. Goat
471. Goat
472. Goat
473. Goat
474. Car
475. Goat
476. Goat
477. Goat
478. Car
479. Goat
480. Goat
481. Goat
482. Car
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Goat
489. Car
490. Car
491. Goat
492. Car
493. Goat
494. Goat
495. Car
496. Goat
497. Goat
498. Goat
499. Car
500. Car
501. Goat
502. Goat
503. Goat
504. Goat
505. Car
506. Goat
507. Car
508. Car
509. Goat
510. Car
511. Car
512. Car
513. Car
514. Car
515. Goat
516. Goat
517. Goat
518. Car
519. Car
520. Goat
521. Goat
522. Car
523. Goat
524. Goat
525. Goat
526. Goat
527. Goat
528. Goat
529. Goat
530. Car
531. Car
532. Car
533. Goat
534. Car
535. Car
536. Goat
537. Goat
538. Goat
539. Goat
540. Goat
541. Car
542. Car
543. Goat
544. Car
545. Goat
546. Goat
547. Goat
548. Car
549. Car
550. Goat
551. Goat
552. Goat
553. Goat
554. Goat
555. Goat
556. Car
557. Goat
558. Car
559. Goat
560. Car
561. Car
562. Car
563. Car
564. Goat
565. Goat
566. Goat
567. Car
568. Goat
569. Goat
570. Goat
571. Car
572. Goat
573. Car
574. Car
575. Car
576. Car
577. Goat
578. Car
579. Goat
580. Car
581. Goat
582. Goat
583. Goat
584. Car
585. Goat
586. Car
587. Goat
588. Goat
589. Car
590. Goat
591. Car
592. Car
593. Car
594. Goat
595. Goat
596. Goat
597. Goat
598. Goat
599. Car
600. Goat
601. Goat
602. Car
603. Car
604. Goat
605. Goat
606. Car
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Goat
613. Car
614. Goat
615. Goat
616. Car
617. Goat
618. Car
619. Goat
620. Goat
621. Car
622. Goat
623. Car
624. Car
625. Goat
626. Car
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Goat
635. Goat
636. Car
637. Goat
638. Car
639. Goat
640. Goat
641. Goat
642. Goat
643. Goat
644. Goat
645. Goat
646. Car
647. Goat
648. Goat
649. Goat
650. Car
651. Goat
652. Goat
653. Car
654. Goat
655. Goat
656. Goat
657. Goat
658. Goat
659. Goat
660. Goat
661. Car
662. Goat
663. Goat
664. Car
665. Car
666. Car
667. Goat
668. Goat
669. Goat
670. Car
671. Car
672. Car
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Goat
680. Goat
681. Goat
682. Goat
683. Goat
684. Car
685. Goat
686. Goat
687. Goat
688. Goat
689. Goat
690. Goat
691. Car
692. Car
693. Car
694. Goat
695. Goat
696. Goat
697. Car
698. Goat
699. Car
700. Goat
701. Car
702. Goat
703. Goat
704. Goat
705. Car
706. Goat
707. Goat
708. Goat
709. Goat
710. Car
711. Goat
712. Goat
713. Goat
714. Goat
715. Goat
716. Car
717. Goat
718. Goat
719. Goat
720. Car
721. Car
722. Goat
723. Goat
724. Car
725. Car
726. Goat
727. Car
728. Goat
729. Goat
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Car
737. Goat
738. Goat
739. Goat
740. Goat
741. Car
742. Goat
743. Goat
744. Goat
745. Car
746. Car
747. Goat
748. Goat
749. Car
750. Goat
751. Goat
752. Goat
753. Goat
754. Car
755. Car
756. Car
757. Car
758. Goat
759. Goat
760. Goat
761. Goat
762. Goat
763. Goat
764. Goat
765. Car
766. Goat
767. Goat
768. Goat
769. Goat
770. Car
771. Car
772. Goat
773. Goat
774. Car
775. Car
776. Goat
777. Goat
778. Car
779. Goat
780. Goat
781. Car
782. Goat
783. Car
784. Goat
785. Car
786. Car
787. Car
788. Goat
789. Car
790. Goat
791. Goat
792. Car
793. Goat
794. Car
795. Goat
796. Goat
797. Goat
798. Car
799. Goat
800. Goat
801. Car
802. Goat
803. Goat
804. Goat
805. Car
806. Car
807. Goat
808. Car
809. Goat
810. Goat
811. Car
812. Goat
813. Car
814. Goat
815. Car
816. Goat
817. Car
818. Goat
819. Car
820. Goat
821. Goat
822. Goat
823. Car
824. Car
825. Car
826. Car
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Goat
835. Goat
836. Goat
837. Goat
838. Car
839. Car
840. Goat
841. Car
842. Car
843. Car
844. Goat
845. Goat
846. Car
847. Goat
848. Goat
849. Goat
850. Goat
851. Goat
852. Car
853. Car
854. Goat
855. Goat
856. Car
857. Car
858. Car
859. Car
860. Goat
861. Goat
862. Goat
863. Car
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Car
873. Goat
874. Goat
875. Goat
876. Goat
877. Goat
878. Goat
879. Car
880. Car
881. Goat
882. Car
883. Car
884. Car
885. Goat
886. Goat
887. Car
888. Goat
889. Goat
890. Goat
891. Goat
892. Goat
893. Car
894. Car
895. Goat
896. Goat
897. Goat
898. Goat
899. Car
900. Goat
901. Goat
902. Goat
903. Goat
904. Car
905. Car
906. Car
907. Goat
908. Goat
909. Car
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Goat
919. Goat
920. Goat
921. Goat
922. Goat
923. Car
924. Goat
925. Car
926. Car
927. Goat
928. Goat
929. Car
930. Car
931. Goat
932. Car
933. Goat
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Goat
940. Car
941. Goat
942. Goat
943. Car
944. Goat
945. Car
946. Car
947. Goat
948. Goat
949. Car
950. Car
951. Goat
952. Goat
953. Goat
954. Goat
955. Goat
956. Car
957. Goat
958. Goat
959. Car
960. Car
961. Goat
962. Goat
963. Goat
964. Goat
965. Goat
966. Car
967. Car
968. Goat
969. Car
970. Goat
971. Goat
972. Car
973. Car
974. Goat
975. Car
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Goat
985. Car
986. Car
987. Goat
988. Goat
989. Goat
990. Goat
991. Car
992. Car
993. Goat
994. Car
995. Goat
996. Goat
997. Goat
998. Goat
999. Car
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 104 times and lost 896 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 35 times and lost 65 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}