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. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
10. Goat

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 347 times and lost 653 times.

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

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 114 times and lost 886 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 31 times and lost 69 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;
	}
}